Distance-regular graphs without 4-claws

TL;DR

This paper classifies certain distance-regular graphs with diameter at least 3 that do not contain a specific type of induced subgraph, expanding understanding of their structural properties.

Contribution

It provides a complete characterization of distance-regular graphs with diameter ≥ 3 and no induced K_{1,4} subgraphs, under the condition c_2 ≥ 2.

Findings

Identifies all such graphs with the specified properties.

Establishes structural constraints for these graphs.

Contributes to the classification theory of distance-regular graphs.

Abstract

We determine the distance-regular graphs with diameter at least $3$ and $c_2\geq 2$ but without induced $K_{1,4}$-subgraphs.

Equations

$$|C|\le 1+\frac{k}{|{\theta }_D|},$$

$$a_1+1>(-2{\theta }_D-1)(c_2-1).$$

$$\iota (\Gamma ):=\{b_0,b_1,\dots ,b_{D-1};c_1,c_2,\dots ,c_D\}$$

$$k_i=\frac{b_1\cdots b_{i-1}}{c_2\cdots c_i}k(2\le i\le D),$$

$$L_{1}:=\left\lgroup\begin{tabular}[]{llllllll}$0$&$k$\\ $c_{1}$&$a_{1}$&$b_{1}$\\ &$c_{2}$&$a_{2}$&$b_{2}$\\ &&.&.&.\\ &&&$c_{i}$&$a_{i}$&$b_{i}$\\ &&&&.&.&.\\ &&&&&$c_{D-1}$&$a_{D-1}$&$b_{D-1}$\\ &&&&&\makebox{\hskip 9.21883pt}&$c_{D}$&$a_{D}$\end{tabular}\right\rgroup.$$

$$c_i{u}_{i-1}(\theta )+a_i{u}_i(\theta )+b_i{u}_{i+1}(\theta )=\theta u_i(\theta )(1\le i\le D),$$

$$m(\theta )=\frac{v}{{\sum }_{i=0}^D{k}_i{u}_i^2(\theta )},$$

$$\sum_{i=0}^{D}m_{i}\theta_{i}^{\ell}=tr(A^{\ell})=\mbox{the~{}number~{}of~{}closed~{}walks~{}of~{}length~{}$\ell$~{}in~{}$\Gamma$}~{}~{}(\ell\geq 1),$$

$$k\ge 3a_1-3c_2+7.$$

$$\left|{\Gamma }_1(x,z_3)\setminus (X_1\cup X_2)\right|\ge a_1-2(c_2-1).$$

$$k=|{\Gamma }_1(x)|\ge |\{z_3\}|+|X_1\cup X_2|+|{\Gamma }_1(x,z_3)\setminus (X_1\cup X_2)|\ge 3a_1-3c_2+7,$$

$$({\theta }_1+1)({\theta }_D+1)\le -b_1,$$

$$m_1\ge c.$$

$$m_1\ge \frac{k}{2}.$$

$${\theta }_D\ge -\frac{t\delta }{2\delta -t}.$$

$$S_i:={\Gamma }_1(x,y_i)\setminus \bigcup _{j\in \{1,2,\dots ,t\}\setminus \{i\}}{\Gamma }_1(x,y_j).$$

$$2(k-t)-|S|=|S|+2|{\Gamma }_1(x)\setminus (Y\cup S)|\le ϵ=t{a}_1=t\left(\frac{k}{\delta }-1\right)$$

$$\left(\frac{2\delta -t}{t\delta }\right)k+1\le |S_1\cup \{x,y_1\}|\le 1+\frac{k}{|{\theta }_D|}$$

$${\theta }_D\ge -\frac{t\delta }{2\delta -t},$$

$$\delta (a_1+1)=k\ge \sum _{i=1}^t|{\Gamma }_1(x,y_i)\cup \{y_i\}|-\sum _{1\le i<j\le t}|{\Gamma }_1(x,y_i,y_j)|\ge t(a_1+1)-\frac{t(t-1)}{2}(c_2-1).$$

$$k$$

$$c_3>2c_2.$$

$${\theta }_0=k,{\theta }_2=-1,{\theta }_i=\frac{k-2c_2-1+(-1{)}^{\frac{i-1}{2}}\sqrt{(k-2c_2-1{)}^2+4k}}{2}(i=1,3),$$

$$m_0=1,m_1=\frac{k(k+1)}{{\theta }_1^2+k},m_2=k,m_3=\frac{{\theta }_1^2(k+1)}{{\theta }_1^2+k}.$$

$${\eta }_1+{\eta }_2=\frac{1}{2}{a}_1-\frac{1}{2}{c}_2-1\text{mbox}and{\eta }_1{\eta }_2=-\frac{1}{2}{a}_1.$$

$$k-\frac{3}{2}{a}_1-1\le 1+\frac{k}{|{\theta }_3|}=1+{\theta }_1.$$

$$\frac{k-1}{4}=k-\frac{3}{2}{a}_1-1\le 1+{\theta }_1=1+\sqrt{k}$$

$$-1-\frac{b_1}{{\theta }_1+1}=-\ell -1$$

$$c_2=\frac{\ell }{\ell +1}{a}_1+2\ell \text{mbox}andthusk=(2\ell +1)\left(\frac{a_1}{\ell +1}+1\right).$$

$${\psi }_i(x,C):=\left|\{z\in C\mid \partial (x,z)=i\}\right|.$$

Full text

1 Introduction

A $t$-claw is a complete bipartite graph $K_{1,t}$ with parts of size $1$ and $t$. We say that a graph is $t$-claw-free (or without $t$-claws), if it does not contain a $t$-claw as an induced subgraph. For further definitions and notations, we refer the reader to Section 2.

Much attention has been paid to the study of $3$-claw-free graphs as a generalization of line graphs (it is easy to see that line graphs are $3$-claw-free). The most general result in this direction belongs to Chudnovsky and Seymour [18], who showed in a series of papers that every connected $3$-claw-free graph can be obtained from one of the so-called basic $3$-claw-free graphs by some simple operations. Kabanov and Makhnev [25] and (independently) Blokhuis and Brouwer [11] determined all distance-regular graphs without 3-claws.

It is a well-known fact that the smallest eigenvalue of a line graph is at least $-2$. In fact, any connected graph has smallest eigenvalue at most $-1$ with equality if and only if the graph is complete, i.e., its vertices are pairwise adjacent. It was shown by Cameron et al. [17] (cf. [16, Theorem 3.12.2]) that a connected regular graph with smallest eigenvalue at least $-2$ is either a line graph or a cocktail party graph, or the number of its vertices is at most 28.

We recall that a clique in a graph is a set of vertices that induces a complete subgraph. If a graph $\Gamma$ is regular with valency $k$, then its line graph ${\cal L}(\Gamma)$ is regular with valency $2(k-1)$, and it contains a set ${\cal C}$ of maximal cliques of the same order equal to $1+\frac{2(k-1)}{2}$ such that every edge of ${\cal L}(\Gamma)$ belongs to a unique member of ${\cal C}$.

For distance-regular graphs, this observation can be formulated in a more general way. Let $\Gamma$ be a distance-regular graph with valency $k$, diameter $D\geq 2$ and smallest eigenvalue $\theta_{D}$. Then any clique $C$ of $\Gamma$ satisfies the inequality

[TABLE]

which is known as the Delsarte bound (see [16, Proposition 4.4.6]). This was shown by Delsarte [20] for strongly regular graphs, and Godsil [22] generalised it to distance-regular graphs. A clique $C$ in $\Gamma$ is called a Delsarte clique, if $C$ contains exactly $1+\frac{k}{|\theta_{D}|}$ vertices. Note that, if $\Gamma$ contains a Delsarte clique, then $\theta_{D}$ must be integral as $1+\frac{k}{|\theta_{D}|}$ is integral.

Godsil [22] introduced the following notion of a geometric distance-regular graph: a non-complete distance-regular graph $\Gamma$ is called geometric, if there exists a set of Delsarte cliques ${\cal C}$ such that every edge of $\Gamma$ lies in a unique member of ${\cal C}$. In this case we say that $\Gamma$ is geometric with respect to $\mathcal{C}$. In case of diameter 2, this notion is equivalent to that of geometric strongly regular graphs as introduced by Bose [14]. Examples of geometric distance-regular graphs include the Hamming graphs, the Johnson graphs, the Grassmann graphs, the dual polar graphs, the bilinear forms graphs.

Note that the notion of a geometric distance-regular graph is stronger than those of a clique geometry, as defined by Metsch [31], or an asymptotic Delsarte geometry, studied by Babai and Wilmes [3]. Both definitions played an important role in recent progress on the complexity of the graph isomorphism problem (restricted to the class of strongly regular graphs) [2], and on the problem of classifying primitive coherent configurations with large automorphism groups [36]. In particular, Spielman [35] improved the complexity of isomorphism testing of strongly regular graphs, found by Babai [1], using the following result by Neumaier [32]: for a fixed integer $\theta\geq 2$, there are only finitely many coconnected non-geometric distance-regular graphs with smallest eigenvalue at least $-\theta$ and diameter $2$. Koolen and Bang [27] generalised this as follows: there are only finitely many coconnected non-geometric distance-regular graphs with smallest eigenvalue at least $-\theta$ and with given diameter $D\geq 2$ or having the intersection number $c_{2}\geq 2$ (in fact, the Bannai-Ito conjecture recently proved by Bang et al. [5] shows that the latter condition on $c_{2}$ is not necessary).

Neumaier [32] also showed that except for a finite number of graphs, all coconnected strongly regular graphs with a given smallest eigenvalue are geometric, and they are either Latin square graphs or Steiner graphs. In [37], Wilson showed that there are super-exponentially many Steiner graphs, and similarly there are super-exponentially many Latin square graphs for certain parameter sets, see. This shows that the above-mentioned result of Neumaier is the best we can hope for the case of distance-regular graphs of diameter two. The situation for distance-regular graphs with larger diameter seems to be different: Koolen and Bang [27] conjectured that, for a fixed integer $\theta\geq 2$, any geometric distance-regular graph with smallest eigenvalue $-\theta$, diameter $D\geq 3$ and $c_{2}\geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph or the number of vertices is bounded above by a function of $\theta$.

Note that a geometric distance-regular graph with smallest eigenvalue $-t$ is $(t+1)$-claw-free, since every edge is contained in exactly one Delsarte clique, and hence each vertex is contained in exactly $t$ Delsarte cliques.

On the other hand, Godsil [22] showed that $\Gamma$ is geometric, if $\Gamma$ is $t$-claw-free for any $t>|\theta_{D}|$ and the intersection numbers $a_{1}$ and $c_{2}$ of $\Gamma$ satisfy the following inequality:

[TABLE]

Koolen and Bang [27] showed that $\Gamma$ is geometric, if $a_{1}\geq\lfloor\theta_{D}\rfloor^{2}c_{2}$ holds. Moreover, they [19, Proposition 9.7] showed that a distance-regular graph $\Gamma$ of diameter $D\geq 2$ having valency $k\geq{\rm max}\{t^{2}-t,\frac{t^{2}-1}{t}(a_{1}+1)\}$ for some integer $t\geq 3$, is geometric with smallest eigenvalue $-t$ if and only if $\Gamma$ is $(t+1)$-claw-free.

Clearly, a connected graph without 2-claws is a complete graph. As we mentioned above, distance-regular graphs without 3-claws were classified in [25] and [11].

In this paper, we classify distance-regular graphs without $4$-claws. Hiraki, Nomura and Suzuki [24] and Yamazaki [38] considered distance-regular graphs that are locally a disjoint union of three cliques of size $a_{1}+1$ (i.e., 4-claw-free), and these graphs for $a_{1}\geq 1$ turn out to be geometric with smallest eigenvalue $-3$ (see Remark 4.6). The problem of classification of such graphs seems to be very hard. Hence we may assume $c_{2}\geq 2$. Bang [4] showed that if $k>\frac{8}{3}(a_{1}+1)$, then a distance-regular graph of valency $k$ is $4$-claw-free if and only if $\Gamma$ is geometric with smallest eigenvalue $-3$. In [4], Bang started the classification of geometric distance-regular graphs with smallest eigenvalue $-3$ and $c_{2}>1$, and the results of Bang and Koolen [7] and Gavrilyuk and Makhnev [21] completed the classification.

Our main result is the following classification of $4$-claw-free distance-regular graphs.

Theorem 1.1

Let $\Gamma$ be a distance-regular graph with diameter $D\geq 3$, valency $k>3$ but without $4$-claws. If $c_{2}\geq 2$, then $\Gamma$ is one of the following graphs:

• (1)

The Taylor graph with $\iota(\Gamma)=\{k,\frac{k-1}{2},1;1,\frac{k-1}{2},k\}$ with $k\in\{5,13,17\}$;

• (2)

The Klein graph with $\iota(\Gamma)=\{7,4,1;1,2,7\}$;

• (3)

The Gosset graph with $\iota(\Gamma)=\{27,10,1;1,10,27\}$;

• (4)

The halved $e$-cube with $\iota(\Gamma)=\{{e\choose 2},{e-2\choose 2},1;1,6,15\}$ with $e\in\{6,7\}$;

• (5)

The Hamming graph $H(3,q)$ with $q\geq 3$;

• (6)

The Johnson graph $J(n,3)$ with $n\geq 6$;

or $\Gamma$ is a putative distance-regular graph with one of the following intersection arrays:

• (7)

$\iota(\Gamma)=\{44,24,1;1,12,44\}$;

• (8)

$\iota(\Gamma)=\{64,34,1;1,17,64\}$;

• (9)

$\iota(\Gamma)=\{104,54,1;1,27,104\}$.

The proof of Theorem 1.1 is organised according to the following three cases with respect to the valency $k$ and the intersection number $a_{1}$ of a distance-regular graph $\Gamma$ satisfying the condition of the theorem:

• (T)

$k\leq 2(a_{1}+1)$,

• (N)

$2(a_{1}+1)<k\leq\frac{8}{3}(a_{1}+1)$,

• (G)

$k>\frac{8}{3}(a_{1}+1)$.

If $\Gamma$ satisfies case (T), then it is either the Johnson graph $J(7,3)$ or the halved $7$-cube or a Taylor graph (see Proposition 2.1). In Section 3, we classify the Taylor graphs without $4$-claws.

Now we consider the distance-regular graphs that satisfy cases (G) or (N). Here we show that the diameter $D$ of $\Gamma$ is at most 4 (see Lemma 2.10), and the smallest eigenvalue $\theta_{D}$ of $\Gamma$ is bounded below in terms of the ratio $\frac{k}{a_{1}+1}$ (see Lemma 2.7).

If $\Gamma$ satisfies case (G), then it was essentially shown by Bang (see Proposition 2.11) that $\Gamma$ is $4$-claw-free if and only if it is geometric with smallest eigenvalue $-3$. If $\Gamma$ is geometric with case (N), then we observe that there are only finitely many possibilities for $\theta_{D}$, and this enables us to classify all geometric distance-regular graphs with $c_{2}\geq 2$ but without $4$-claws in Section 4.

In Section 5, we consider non-geometric distance-regular graphs with case (N). Since the smallest eigenvalue $\theta_{D}$ of $\Gamma$ is bounded below (see Lemma 2.7), it follows by [27] that there are only finitely many non-geometric distance-regular graphs. In particular, they showed that the valency $k$ is bounded above as $O(m_{1}^{2})$, where $m_{1}$ is the multiplicity of second largest eigenvalue. However, their bound for $k$ is too large, which makes the search for feasible intersection arrays for $\Gamma$ almost intractable. In Subsection 5.1, we obtain a better valency bound of order $O(m_{1})$ by using that any clique size gives a lower bound for $m_{1}$ (see Lemma 2.6), while the size of a maximum clique in its turn can be bounded below by using the 4-claw-free property (see Lemma 2.7). In Subsection 5.2, we find all feasible intersection arrays, whose valencies satisfy the bounds of Subsection 5.1. For most of the arrays we found, we show in Subsection 5.3 that there are no corresponding distance-regular graphs without $4$-claws. Moreover, in Proposition 5.20, we show that there does not exist a distance-regular graph with intersection array $\{39,24,1;1,4,39\}$.

Some of our results can be possibly generalised for distance-regular graphs without $t$-claws for any $t$, and Section 2 contains some basic definitions from the theory of distance-regular graphs and preliminary results on intersection numbers and eigenvalues of a distance-regular graph without $t$-claws.

Note that the complement of a $t$-claw-free strongly regular graph does not contain any $t$-clique. In general, it appears to be hard to show the non-existence of $t$-cliques in a strongly regular graph by using only its parameters (for example, the Shrikhande graph and the $(4\times 4)$-grid both have the same parameters, but the latter one contains a $4$-clique, while the former one does not; see further [12] for the case $t=4$). This suggests that the assumption $D\geq 3$ in Theorem 1.1 is crucial.

2 Definitions and preliminaries

2.1 Definitions and notation

All graphs considered in this paper are finite, undirected and simple (for more background information, see [16]). For a connected graph $\Gamma=(V(\Gamma),E(\Gamma))$, the distance $\partial(x,y)$ between any two vertices $x,y$ is the length of a shortest path between $x$ and $y$ in $\Gamma$, and the diameter $D$ is the maximum distance ranging over all pairs of vertices of $\Gamma$.

For any vertex $x$ of $\Gamma$, let $\Gamma_{i}(x)$ be the set of vertices in $\Gamma$ at distance precisely $i$ from $x$, where $0\leq i\leq D$. For a non-empty subset $S\subseteq V(\Gamma)$, $\langle S\rangle$ denotes the induced subgraph on $S$. The local graph of a vertex $x$ is $\langle\Gamma_{1}(x)\rangle$. For a class $\mathcal{G}$ of graphs, a graph $\Gamma$ is called locally $\mathcal{G}$, if, for any vertex of $V(\Gamma)$, its local graph is isomorphic to a graph from $\mathcal{G}$. If all graphs from $\mathcal{G}$ are isomorphic to a graph $\Delta$, we say that $\Gamma$ is locally $\Delta$. A graph $\Gamma$ is regular with valency $k$, if $\Gamma_{1}(x)$ contains precisely $k$ vertices for all $x\in V(\Gamma)$.

For a set of vertices $x_{1},\ldots,x_{n}$ of $\Gamma$, let $\Gamma_{1}(x_{1},\ldots,x_{n})$ denote $\cap_{i=1}^{n}\Gamma_{1}(x_{i})$. Recall that a clique (coclique, resp.) is a set of pairwise adjacent (non-adjacent, resp.) vertices. A connected non-complete graph $\Gamma$ is called Terwilliger, if for any two vertices $x,y$ at distance $2$, $\langle\Gamma_{1}(x,y)\rangle$ is a clique, whose order does not depend on the choice of $x$ and $y$.

A connected graph $\Gamma$ with diameter $D$ is called distance-regular,  if there exist integers $b_{i-1}$, $c_{i}$ $(1\leq i\leq D$) such that, for any two vertices $x,y\in V(\Gamma)$ with $\partial(x,y)=i$, there are precisely $c_{i}$ neighbors of $y$ in $\Gamma_{i-1}(x)$ and $b_{i}$ neighbors of $y$ in $\Gamma_{i+1}(x)$. Set $c_{0}=b_{D}=0$. The numbers $b_{i},c_{i}$ and $a_{i}:=b_{0}-b_{i}-c_{i}~{}(0\leq i\leq D)$ are called the intersection numbers of $\Gamma$. We observe $a_{0}=0$ and $c_{1}=1$. The array

[TABLE]

is called the intersection array of $\Gamma$. In particular, $\Gamma$ is a regular graph with valency $k:=b_{0}$. We further note that, for $1\leq i\leq D$, the number $|\Gamma_{i}(x)|$ does not depend on the choice of a vertex $x$ of $\Gamma$. Define $k_{i}:=|\Gamma_{i}(x)|$ for any vertex $x$ and $i=0,1,\ldots,D$. Then we have $k_{0}=1,~{}k_{1}=k,~{}c_{i+1}k_{i+1}=b_{i}k_{i}~{}~{}(0\leq i\leq D-1)$ and thus

[TABLE]

so that $v:=|V(\Gamma)|=k_{0}+k_{1}+\ldots+k_{D}$.

A distance-regular graph with diameter 2 is called a strongly regular graph. We say that a strongly regular graph $\Gamma$ has parameters $(v,k,\lambda,\mu)$, where $\lambda:=a_{1}$, and $\mu:=c_{2}$.

Let $\Gamma$ be a distance-regular graph of diameter $D\geq 2$. The adjacency matrix $A:=A(\Gamma)$ is the $v\times v$-matrix with rows and columns indexed by $V(\Gamma)$, where the $(x,y)$-entry of $A$ is $1$, if $\partial(x,y)=1$, and [math] otherwise. The eigenvalues of $\Gamma$ are the eigenvalues of $A$. It is well-known that a distance-regular graph $\Gamma$ with diameter $D$ has exactly $D+1$ distinct eigenvalues $k=\theta_{0}>\theta_{1}>\cdots>\theta_{D}$, which are the eigenvalues of the following tridiagonal matrix (cf. [16, p.128]):

[TABLE]

The standard sequence $\{u_{i}(\theta)\mid 0\leq i\leq D\}$ corresponding to an eigenvalue $\theta$ is a sequence satisfying the following recurrence relation:

[TABLE]

where $u_{0}(\theta)=1$ and $u_{1}(\theta)=\frac{\theta}{k}$. Then the multiplicity of eigenvalue $\theta$ is given by

[TABLE]

which is known as Biggs’ formula (cf. [9, Theorem 21.4], [16, Theorem 4.1.4]). Let $m_{i}$ denote the multiplicity of eigenvalue $\theta_{i}~{}(0\leq i\leq D)$. Then

[TABLE]

where $tr(A^{\ell})$ is the trace of matrix $A^{\ell}$ (cf. [9, Lemma 2.5]).

Let $\mathbb{Z}$ and $\mathbb{N}$ denote the set of integers and the set of positive integers, respectively.

2.2 Basic results

In this subsection, we collect several basic results on distance-regular graphs. For the rest of Section 2, let $\Gamma$ be a distance-regular graph of diameter $D\geq 2$, and with intersection array given by Eq. (2), valency $k$, and $D+1$ distinct eigenvalues $\theta_{0}=k$ $>$ $\theta_{1}$ $>$ $\ldots$ $>$ $\theta_{D}$ with multiplicities $m_{0}=1$, $m_{1}$, $\ldots$, $m_{D}$, respectively.

Proposition 2.1

([28, Theorem 16]) If $D\geq 3$ and $k\leq 2(a_{1}+1)$, then $\Gamma$ is one of the following graphs:

• (1)

A polygon;

• (2)

The line graph of a Moore graph;

• (3)

The flag graph of a regular generalized $D$-gon of order $(s,s)$ for some $s$;

• (4)

A Taylor graph;

• (5)

The Johnson graph $J(7,3)$;

• (6)

The halved $7$-cube.

Lemma 2.2

If $k>2(a_{1}+1)$ and $\Gamma$ is not a Terwilliger graph, then $c_{i}-b_{i}\geq c_{i-1}-b_{i-1}+a_{1}+2$ holds (for all $1\leq i\leq D$) and

[TABLE]

*Proof: *The inequality on $c_{i}-b_{i}$ follows from [16, Theorem 5.2.1]. Suppose that $\langle\{x,z_{1},y,z_{2}\}\rangle$ is an induced quadrangle where $\partial(x,y)=\partial(z_{1},z_{2})=2$. Put $X_{i}:=\{z_{i}\}\cup\Gamma_{1}(x,z_{i})$ for $i=1,2$. Then $2(a_{1}+1)-(c_{2}-2)\leq|X_{1}\cup X_{2}|\leq 2(a_{1}+1)$. As $k>2(a_{1}+1)$, there exists a vertex $z_{3}\in\Gamma_{1}(x)\setminus(X_{1}\cup X_{2})$ and we find

[TABLE]

Hence

[TABLE]

which shows the result.

The following results relate eigenvalues and intersection numbers of $\Gamma$.

Theorem 2.3

([30, Theorem 3.6]) The following inequality holds:

[TABLE]

with equality if and only if $D=2$.

Lemma 2.4

([30, Proposition 3.2]) If $D=3$, then the following holds.

• (1)

$\theta_{1}>\max\{a_{1}+1-c_{2},a_{3}-b_{2}\}$.

• (2)

$\theta_{1}\geq\min\{a_{3},\frac{a_{1}+\sqrt{a_{1}^{2}+4k}}{2}\}$.

• (3)

$\theta_{2}$* lies between $-1$ and $a_{3}-b_{2}$.*

Lemma 2.5

*If $\theta_{1}<a_{1}$, then $D\leq 3$, and $D=3$ implies $b_{1}<c_{3}$. *

*Proof: *(1) Suppose $D\geq 4$. For any two vertices $x,y$ with $\partial(x,y)=4$, the subgraph $\langle\Gamma_{1}(x)\cup\Gamma_{1}(y)\rangle$ is the disjoint union of two connected regular graphs with valency $a_{1}$. By the interlacing result (see [16, Theorem 3.3.1]) we have $\theta_{0}\geq a_{1}$ and $\theta_{1}\geq a_{1}$, a contradiction.

(2) By (1) and Lemma 2.4 (2), we find that $D=3$ and $a_{3}\leq\theta_{1}<a_{1}$. Hence the result follows by $b_{1}=k-2-(a_{1}-1)\leq k-2-a_{3}=c_{3}-2<c_{3}$.

The following result gives some bounds on the multiplicity of the second largest eigenvalue of $\Gamma$.

Lemma 2.6

(cf. [34, Theorem 3], [16, Theorem 4.4.4]) Suppose that $D\geq 3$ holds.

• (1)

If $\Gamma$ has a clique of size $c$, then

[TABLE]

• (2)

If $\theta_{1}$ is irrational with multiplicity $m_{1}<k$, then

[TABLE]

*Proof: *(1) It is straightforward by [34, Theorem 3].

(2) By [16, Theorem 4.4.4], $\theta_{1}+1$ and $\theta_{D}+1$ are conjugate and thus $m_{1}=m_{D}$. Moreover, each local graph of $\Gamma$ has eigenvalues $-1-\frac{b_{1}}{\theta_{1}+1}$ and $-1-\frac{b_{1}}{\theta_{D}+1}$ with multiplicities at least $k-m_{1}$ and $k-m_{D}$, respectively. This shows that $k\geq(k-m_{1})+(k-m_{D})=2k-2m_{1}$, and hence we find $m_{1}\geq\frac{1}{2}k$.

2.3 Some structure theory for distance-regular graphs without $(t+1)$-claws

In this subsection, we give some results on distance-regular graphs without $(t+1)$-claws ($t\geq 3$).

Lemma 2.7

Suppose that $\Gamma$ contains a $t$-claw but no $(t+1)$-claws for some integer $t\geq 3$. With $\delta:=\frac{k}{a_{1}+1}$, the following holds.

• (1)

$\Gamma$* contains a clique with at least $\left(\frac{2\delta-t}{t\delta}\right)k+1$ vertices. In particular,*

[TABLE]

• (2)

$c_{2}\geq 1+\frac{2(t-\delta)}{t(t-1)}(a_{1}+1)$.

*Proof: *By assumption, $\Gamma$ contains a $t$-claw, say $\langle\{x,y_{1},\ldots,y_{t}\}\rangle$, where $\partial(x,y_{i})=1~{}(1\leq i\leq t)$.

(1) For each $1\leq i\leq t$, we define a set

[TABLE]

We first show the following claim.

Claim 2.8

There is an integer $1\leq j\leq t$ such that $|S_{j}|\geq\left(\frac{2\delta-t}{t\delta}\right)k-1$.

*Proof of Claim 2.8: * Let $\epsilon$ be the number of edges between $Y:=\{y_{1},\ldots,y_{t}\}$ and $\Gamma_{1}(x)\setminus Y$. As each vertex in $Y$ has exactly $a_{1}$ neighbors in $\Gamma_{1}(x)$, we have $\epsilon=ta_{1}$. Since each vertex in $S:=\cup_{i\in\{1,\ldots,t\}}S_{i}$ has a unique neighbor in $Y$ and each vertex in $\Gamma_{1}(x)\setminus(Y\cup S)$ has at least two neighbors in $Y$, we find $\epsilon\geq|S|+2|\Gamma_{1}(x)\setminus(Y\cup S)|$. Hence it follows by $a_{1}=\frac{k}{\delta}-1$ and $|S|+|\Gamma_{1}(x)\setminus(Y\cup S)|=|\Gamma_{1}(x)\setminus Y|=k-t$ that

[TABLE]

and thus $|S|\geq\left(2-\frac{t}{\delta}\right)k-t$. Now the claim follows as $|S|=\sum_{i=1}^{t}|S_{i}|\geq\left(2-\frac{t}{\delta}\right)k-t$.

Without loss of generality, let $S_{1}$ satisfy $|S_{1}|\geq\left(\frac{2\delta-t}{t\delta}\right)k-1$ by Claim 2.8. If there exist two non-adjacent vertices $v_{1},v_{2}\in S_{1}$, then $\langle\{x,v_{1},v_{2},y_{2},\ldots,y_{t}\}\rangle$ is a $(t+1)$-claw, which is impossible. So $S_{1}\cup\{x,y_{1}\}$ induces a clique and thus

[TABLE]

follows by Eq. (1). Now we have

[TABLE]

and this shows (1).

(2) Since there are no $(t+1)$-claws in $\Gamma$, we have $\Gamma_{1}(x)=\cup_{1\leq i\leq t}\left(\Gamma_{1}(x,y_{i})\cup\{y_{i}\}\right)$. It follows by the principle of inclusion and exclusion that

[TABLE]

This completes the proof.

In the following result, we find a slightly better bound for clique size when $t=3$. Define $\mu_{\min}$ by $\mu_{\min}:=\min\{|\Gamma_{1}(x,y,z)|\mid x\in V(\Gamma)\mbox{~{}and~{}}y,z\in\Gamma_{1}(x)\mbox{~{}satisfying~{}}\partial(y,z)=2\}.$

Corollary 2.9

Suppose that $\Gamma$ contains a $3$-claw but no $4$-claws. Then there exists a clique of size at least $k-1-2a_{1}+\mu_{\min}$.

*Proof: *Let $\{x,y_{1},y_{2},y_{3}\}$ be a set, which induces a $3$-claw, where $y_{i}\in\Gamma_{1}(x),~{}i=1,2,3$. Put $\gamma:=|\Gamma_{1}(x,y_{1},y_{2},y_{3})|$. As there are no $4$-claws, $\Gamma_{1}(x)=\cup_{i=1}^{3}(\Gamma_{1}(x,y_{i})\cup\{y_{i}\})$ holds. Thus, $\gamma=k-3(a_{1}+1)+\sum_{1\leq i<j\leq 3}|\Gamma_{1}(x,y_{i},y_{j})|$ follows by

[TABLE]

Since there are no $4$-claws in $\Gamma$, we see that $X:=\Gamma_{1}(x,y_{1})\setminus(\Gamma_{1}(x,y_{2})\cup\Gamma_{1}(x,y_{3}))$ induces a clique, and hence $C:=\{x,y_{1}\}\cup X$ induces a clique of size $k-2a_{1}-1+|\Gamma_{1}(x,y_{2},y_{3})|$, which proves the claim.

Lemma 2.10

Suppose that $\Gamma$ contains a $3$-claw but no $4$-claws. If $D\geq 3$ and $3\leq k<3(a_{1}+1)$, then the following holds.

• (1)

$\Gamma$* contains an induced quadrangle (i.e., $c_{2}\geq 2$). *

• (2)

$b_{2}\leq\min\{a_{1}+1,a_{3}+1,\frac{k}{-\theta_{D}}\}$.

• (3)

$D\leq 4$.

• (4)

If $D=3$, then $\theta_{2}\geq-1$.

*Proof: *(1) As $\Gamma$ contains a $3$-claw and $k<3(a_{1}+1)$, it follows from Lemma 2.7 (2) that $c_{2}\geq 2$. Suppose that $\Gamma$ has no induced quadrangles (i.e., $\Gamma$ is a Terwilliger graph). Then, by [28, Proposition 6 (3)], $\Gamma$ is the icosahedron, the Conway-Smith graph, or the Doro graph. Both the Conway-Smith graph and the Doro graph have $4$-claws, and the icosahedron does not have a $3$-claw, a contradiction.

(2) By (1) there is a quadrangle, say $\{x,z_{1},y,z_{2}\}$, where $\partial(x,y)=2=\partial(z_{1},z_{2})$. As $D\geq 3$ and $\Gamma$ is $4$-claw-free, $\langle\Gamma_{3}(x)\cap\Gamma_{1}(y)\rangle$ must be a clique of size $b_{2}$. This shows that $a_{1}\geq b_{2}-1$ and $a_{3}\geq b_{2}-1$. Since $\langle(\Gamma_{3}(x)\cap\Gamma_{1}(y))\cup\{y\}\rangle$ is a clique of size $b_{2}+1$, the result follows by Eq. (1).

(3) Suppose $D\geq 5$. By (1), there is a quadrangle, say $\langle\{y,s,t,w\}\rangle$ with $\partial(y,t)=\partial(s,w)=2$. As $D\geq 5$, there is a vertex $x$ satisfying $\partial(x,y)=3$ and $\partial(x,t)=5$. As $\{s,w\}$ is a coclique in $\Gamma_{1}(y)$ and $\Gamma$ is $4$-claw-free, the set $C:=\Gamma_{2}(x)\cap\Gamma_{1}(y)$ induces a clique of size $c_{3}$. It follows by (2), Lemma 2.2, and $b_{3}\geq c_{2}$ that $c_{3}\geq c_{2}-b_{2}+a_{1}+2+b_{3}\geq c_{2}+1+b_{3}\geq 2c_{2}+1$ and thus

[TABLE]

Let $y_{1}\in C$ and take $x_{1}\in\Gamma_{1}(x,y_{1})$. As $c_{3}>2c_{2}$, there is a vertex $y_{2}\in C\setminus\Gamma_{1}(x_{1})$, and take a vertex $x_{2}\in\Gamma_{1}(x,y_{2})$. Similarly, there is a vertex $y_{3}\in C\setminus(\Gamma_{1}(x_{1})\cup\Gamma_{1}(x_{2}))$ as $c_{3}>2c_{2}$. Take a vertex $x_{3}\in\Gamma_{1}(x,y_{3})$. As the set $C$ induces a clique, for any $1\leq i<j\leq 3$, two vertices $x_{i}$ and $y_{j}$ has $c_{2}$ common neighbors in $C$ and thus $\{x_{1},x_{2},x_{3}\}$ is a coclique. On the other hand, there exists a vertex $x_{4}\in\Gamma_{1}(x)\cap\Gamma_{4}(y)$ by $b_{3}\geq 1$, and the set $\{x_{i}\mid 1\leq i\leq 4\}$ induces a coclique as $\partial(y,x_{i})=2$ ($i=1,2,3$) and $\partial(y,x_{4})=4$. Hence $\langle\{x,x_{i}\mid 1\leq i\leq 4\}\rangle$ is a $4$-claw in $\Gamma$, which is impossible. This shows $D\leq 4$.

(4) As $a_{3}-b_{2}\geq-1$ holds by (2), the result (4) follows from Lemma 2.4 (3).

Proposition 2.11

(cf. [4, Theorem 3.2]) If $k>\frac{8}{3}(a_{1}+1)$, then the following statements are equivalent.

• (1)

$\Gamma$* is $4$-claw-free.*

• (2)

$\Gamma$* is a geometric distance-regular graph with smallest eigenvalue $-3$.*

*Proof: *If $\frac{8}{3}(a_{1}+1)<k\leq 3$, then $k=3$ and $a_{1}=0$. Hence $\Gamma$ is the cube $H(3,2)$ (see [10]), which satisfies both $(1)$ and $(2)$. If $k>\max\{3,\frac{8}{3}(a_{1}+1)\}$, then the result follows from [4, Theorem 3.2].

3 Taylor graphs without $4$-claws

In this section we classify the Taylor graphs without $4$-claws (Theorem 3.1).

Let $\Gamma$ be a Taylor graph (i.e., a distance-regular graph with diameter three and intersection array $\iota(\Gamma)=\{k,c_{2},1;1,c_{2},k\}$). The eigenvalues $\theta_{i}~{}(0\leq i\leq 3)$ of $\Gamma$ are

[TABLE]

whose multiplicities $m_{i}=m(\theta_{i})~{}(0\leq i\leq 3)$ are given by

[TABLE]

Theorem 3.1

Let $\Gamma$ be a Taylor graph without $4$-claws. Then $\Gamma$ is one of the following graphs:

• (1)

The hexagon;

• (2)

The cube with $\iota(\Gamma)=\{3,2,1;1,2,3\}$;

• (3)

The icosahedron with $\iota(\Gamma)=\{5,2,1;1,2,5\}$;

• (4)

The Johnson graph $J(6,3)$ with $\iota(\Gamma)=\{9,4,1;1,4,9\}$;

• (5)

The Taylor graph with $\iota(\Gamma)=\{13,6,1;1,6,13\}$;

• (6)

The halved $6$-cube with $\iota(\Gamma)=\{15,6,1;1,6,15\}$;

• (7)

The Taylor graph with $\iota(\Gamma)=\{17,8,1;1,8,17\}$;

• (8)

The Gosset graph with $\iota(\Gamma)=\{27,10,1;1,10,27\}$.

Moreover, the hexagon, the cube and the Johnson graph $J(6,3)$ are the only geometric Taylor graphs without $4$-claws.

*Proof: *If $\Gamma$ is $3$-claw-free, then $\Gamma$ is either the hexagon or the icosahedron by [11, Theorem 1.2]. We assume that $\Gamma$ is a Taylor graph with $3$-claws but without $4$-claws. If $k=3$, then $\Gamma$ is the cube (see [16, Theorem 7.5.1]). From now on, we assume $k\geq 4$. If $m_{1}=k$, then it follows by Eq. (5) and Eq. (7) that $\theta_{1}=1$ and $\theta_{3}=-k$, which implies that $\Gamma$ is bipartite. This is impossible as $k\geq 4$ and $a_{1}=0$ (i.e., $\Gamma$ contains $4$-claws). Hence $m_{1}\neq k$. As $m_{1}+m_{3}=k+1$, $m_{1}<k$ holds. For a vertex $x\in V(\Gamma)$, let $\Delta:=\langle\Gamma_{1}(x)\rangle$ be the local graph of $x$. By [16, Theorem 1.5.3], $\Delta$ is a strongly regular graph with parameters $(\nu,\kappa,\lambda,\mu)=\left(k,a_{1},a_{1}-\frac{1}{2}c_{2}-1,\frac{1}{2}a_{1}\right)$ and eigenvalues $\eta_{0}=a_{1},~{}\eta_{1},~{}\eta_{2}$, where $\eta_{2}<0<\eta_{1}$ and

[TABLE]

By Eqs. (1), (6) and Corollary 2.9,

[TABLE]

If $m_{1}=m_{3}$, then $m_{1}=m_{3}=\frac{k+1}{2}$ and $\theta_{1}=-\theta_{3}=\sqrt{k}$ follow by Eqs. (6), (7) and $tr(A)=0$ in Eq. (5). As $|V(\Gamma)|ka_{1}=tr(A^{3})=\sum_{i=0}^{3}m_{i}\theta_{i}^{3}$ holds by Eq. (5), we find $a_{1}=c_{2}=\frac{k-1}{2}$ (thus $k$ is odd) and hence $k\leq 25$ holds as

[TABLE]

by Eq. (9). On the other hand, $a_{1}$ is even and $k$ is a sum of two squares by $\mu=\frac{1}{2}a_{1}\in\mathbb{N}$ and [16, Proposition 1.10.5 (i)]. These restrictions imply $k\in\{5,9,13,17,25\}$. If $k=5$, then $\Gamma$ is the icosahedron, which is $3$-claw-free. If $k=9$, then $\Gamma$ is the Johnson graph $J(6,3)$ which contains $3$-claws but no $4$-claws. If $k=13,17$, then $\Delta$ is the Payley graphs $P(13)$ and $P(17)$, respectively. As the independence numbers of these Payley graphs are all three, $\Gamma$ contains $3$-claws but does not contain $4$-claws. If $k=25$, then each local graph of $\Gamma$ is a strongly regular graph with parameters $(25,12,5,6)$. Since all such stongly regular graphs contain $4$ cocliques (see [33]), $\Gamma$ contains $4$-claws and thus $k\neq 25$. Now the results (4), (5) and (7) follow.

Now we assume that $m_{1}\neq m_{3}$. Then all the eigenvalues $\theta_{i}~{}(0\leq i\leq 3)$ are integral. As $m_{1}<k$ and by [16, Theorem 4.4.4], $b_{1}=\ell(\theta_{1}+1)$ holds for $\ell\in\mathbb{N}$ with $1\leq\ell\leq-\theta_{3}-1$, and

[TABLE]

is the smallest eigenvalue of $\Delta$, i.e., $\eta_{2}=-\ell-1$. It follows by Eq. (8) with $\eta_{2}=-\ell-1$ that

[TABLE]

By Eqs. (9), (10) and (11), we find that $(\ell-3)a_{1}+4(\ell^{2}-1)\leq 0$, which shows $\ell\in\{1,2\}$. If $\ell=1$, then $c_{2}=\frac{k+3}{3}$ and thus $\iota(\Gamma)=\{k,\frac{k+3}{3},1;1,\frac{k+3}{3},k\}$ with $\theta_{1}=\frac{k}{3}$ and $\theta_{3}=-3$ by Eq. (6). As $m_{1}=9-\frac{72}{k+9}\in\mathbb{N}$ holds by Eq. (7), we find $k\in\{15,27,63\}$ as $m_{1}\neq m_{3}$. By [16, Corollary 1.15.3], $k\neq 63$. If $k=15$, then $\Gamma$ is the halved $6$-cube whose local graph is triangular graph $T(6)$ and independence number of $T(6)$ is three. If $k=27$, then $\Gamma$ is the Gosset graph whose local graph is the Schläfli graph and its independence number is also three. If $\ell=2$, then it follows by Eq. (11) that $\iota(\Gamma)=\{k,\frac{2k+10}{5},1;1,\frac{2k+10}{5},k\}$ holds and thus $\theta_{1}=\frac{k}{5}$, $\theta_{3}=-5$ and $m_{1}=25-\frac{600}{k+25}$ by Eqs. (6) and (7). As $m_{1}\in\mathbb{N}$, $m_{1}<k$, $m_{1}\neq m_{3}$ and $a_{1}$ is even, $k\in\{15,35,75,95,125,175,275,575\}$. Note that $\Delta^{c}$ (the complement of local graph $\Delta$) is also a strongly regular graph with parameters $\left(k,\frac{2k+10}{5},\frac{k+25}{10},\frac{k+5}{5}\right)$. It follows by [12, Theorem 3.3] and [13] that for each $k\in\{15,35,75,95,125,175,575\}$, $\Delta^{c}$ contains cliques of size $4$. This implies $\Delta$ contains cocliques of size $4$ and thus $\Gamma$ does contain $4$-claws, a contradiction. If $k=275$, then $\Delta$ is the McLaughlin graph which has a coclique of size $22$, this is also impossible. This completes the proof.

4 Geometric distance-regular graphs without $4$-claws

In this section we classify geometric distance-regular graphs with diameter $D\geq 3$ but without $4$-claws (see Theorem 4.5).

Let $\Gamma$ be a geometric distance-regular graph with respect to a set of Delsarte cliques $\mathcal{C}$ and diameter $D\geq 3$. For a vertex $x\in V(\Gamma)$ and a non-empty subset $S$ of $V(\Gamma)$, we denote $\partial(x,S)=\min\{\partial(x,y)\mid y\in S\}$. For a Delsarte clique $C\in\mathcal{C}$ and for each $i=0,1,\ldots,D-1$, let $C_{i}:=\{x\in V(\Gamma)\mid\partial(x,C)=i\}$. For a vertex $x\in C_{i}$, define

[TABLE]

For $x,y\in V(\Gamma)$ with $\partial(x,y)=j~{}~{}(j=1,2,\ldots,D)$, define

[TABLE]

Since parameters $\psi_{i}(x,C)$ and $\tau_{j}(x,y;\mathcal{C})$ depend only on the distances $i$ and $j$ respectively (see [6, Lemma 4.1] and [23, Section 11.7]), we denote $\psi_{i}:=\psi_{i}(x,C)~{}~{}(i=0,1,\ldots,D-1)$ and $\tau_{j}:=\tau_{j}(x,y;\mathcal{C})~{}(j=1,2,\ldots,D)$. We observe that for any vertex $x\in V(\Gamma)$ and for any Delsarte clique $C\in\mathcal{C}$,

[TABLE]

The next lemmas summarize some results for geometric distance-regular graphs.

Lemma 4.1

([4, Lemma 4.1, Lemma 4.2], [6, Lemma 5.1]) Let $\Gamma$ be a geometric distance-regular graph with diameter $D\geq 2$. Then the following holds.

• (1)

$b_{i}=-(\theta_{D}+\tau_{i})\left(1-\frac{k}{\theta_{D}}-\psi_{i}\right)~{}(1\leq i\leq D-1)$.

• (2)

$c_{i}=\tau_{i}\psi_{i-1}~{}(1\leq i\leq D)$.

• (3)

$c_{2}=\psi_{1}$* if and only if $c_{2}=1$.*

• (4)

If $D\geq 3$, then $\psi_{1}\leq\tau_{2}\leq-\theta_{D}-1$.

Lemma 4.2

([19, Proposition 9.9]) Let $t\geq 2$ be an integer, and let $\Gamma$ be a distance-regular graph with diameter $D\geq 2$ and valency $(t-1)(a_{1}+1)<k<t(a_{1}+t)$. If $a_{1}>\frac{t(t+1)(c_{2}+1)}{2}$, then $\Gamma$ is geometric with $\theta_{D}=-t$.

Lemma 4.3

Let $\Gamma$ be a geometric distance-regular graph with $3$-claws but without $4$-claws. Then either $k=3(a_{1}+1)$ with $c_{2}=1$ or $D\leq 3$ with $c_{2}\geq 2$.

*Proof: *As $\Gamma$ is $4$-claw-free, $3\leq k\leq 3(a_{1}+1)$. It follows by Lemma 2.10 (1) that either $k=3(a_{1}+1)$ with $c_{2}=1$ or $3\leq k<3(a_{1}+1)$ with $c_{2}\geq 2$. Let $\Gamma$ satisfy $3\leq k<3(a_{1}+1)$ and $c_{2}\geq 2$. Then it is enough to show $D\leq 3$. We first show the following claim.

Claim 4.4

For any vertices $x$ and $y$ with $\partial(x,y)=2$, there is a quadrangle containing $x$ and $y$.

Proof of Claim 4.4: Let $\partial(x,y)=2$. As $\Gamma$ is geometric, there is a Delsarte clique $C$ satisfying $x\in C$ and $\partial(y,C)=1$. As $c_{2}\geq 2$, we have $c_{2}>\psi_{1}$ by Lemma 4.1 (3) and thus there exists a vertex $w\in\Gamma_{1}(x,y)\setminus C$. By $|\Gamma_{1}(w)\cap C|=|\Gamma_{1}(y)\cap C|=\psi_{1}$, $x\in\Gamma_{1}(w)\cap C$ and $x\not\in\Gamma_{1}(y)\cap C$, there exists a vertex $z\in\Gamma_{1}(y)\cap C$ such that $\partial(z,w)\neq 1$ and thus $\partial(z,w)=2$. Hence the induced subgraph on $\{x,z,y,w\}$ is a quadrangle.

If $D\geq 4$, then there exist $3$ vertices $p,q,r\in V(\Gamma)$ satisfying $\partial(p,q)=\partial(q,r)=2$ and $\partial(p,r)=4$. By Claim 4.4, there are $4$ vertices $s_{1},s_{2}\in\Gamma_{1}(p,q)$ and $s_{3},s_{4}\in\Gamma_{1}(q,r)$ such that $\langle\{p,s_{1},q,s_{2}\}\rangle$ and $\langle\{q,s_{3},r,s_{4}\}\rangle$ are quadrangles, respectively. This is impossible as the set $\{q,s_{1},s_{2},s_{3},s_{4}\}$ induces a $4$-claw. This shows $D\leq 3$.

Now we prove the main result of this section.

Theorem 4.5

Let $\Gamma$ be a geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$ but without $4$-claws. Then $\Gamma$ is one of the following graphs:

• (1)

One of the two generalized hexagons of order $(2,2)$ with $\iota(\Gamma)=\{6,4,4;1,1,3\}$;

• (2)

The generalized hexagon of order $(8,2)$ with $\iota(\Gamma)=\{24,16,16;1,1,3\}$;

• (3)

The halved Foster graph with $\iota(\Gamma)=\{6,4,2,1;1,1,4,6\}$;

• (4)

A generalized octagon of order $(4,2)$ with $\iota(\Gamma)=\{12,8,8,8;1,1,1,3\}$;

• (5)

The Hamming graph $H(3,q)$ with $q\geq 3$;

• (6)

The Johnson graph $J(n,3)$ with $n\geq 6$;

• (7)

The line graph of a Moore graph;

• (8)

The flag graph of a regular generalized $D$-gon of order $(s,s)$ for some $s$;

• (9)

A distance-regular graph with $\theta_{D}=-3$ satisfying $k=3(a_{1}+1)$, $c_{2}=1$ and $a_{1},\frac{1}{3}c_{D}\geq 2$.

*Proof: *If $\Gamma$ is $3$-claw-free, then it follows by [11, Theorem 1.2] that $\Gamma$ is (7) or (8). (Note that these graphs are all geometric as they satisfy $k=2(a_{1}+1)$ and $\theta_{D}=-2$, and the icosahedron is non-geometric as it has irrational smallest eigenvalue.) Now we suppose that $\Gamma$ contains $3$-claws. If $k\leq 2(a_{1}+1)$, then it follows by Proposition 2.1 and Theorem 3.1 that $\Gamma$ is the Johnson graph $J(n,3)$ with $n\in\{6,7\}$. If $k>\frac{8}{3}(a_{1}+1)$, then it follows by Proposition 2.11, [4, Theorem 4.3] and [7, Theorem 1.4] that $\Gamma$ is one of the following:

(a) One of the two generalized hexagons of order $(2,2)$ with $\iota(\Gamma)=\{6,4,4;1,1,3\}$;

(b) The generalized hexagon of order $(8,2)$ with $\iota(\Gamma)=\{24,16,16;1,1,3\}$;

(c) The halved Foster graph with $\iota(\Gamma)=\{6,4,2,1;1,1,4,6\}$;

(d) A generalized octagon of order $(4,2)$ with $\iota(\Gamma)=\{12,8,8,8;1,1,1,3\}$;

(e) The Hamming graph $H(3,q)$ with $q\geq 3$;

(f) The Johnson graph $J(n,3)$ with $n\geq 20$;

(g) A distance-regular graph with $\theta_{D}=-3$ satisfying $k=3(a_{1}+1)$, $c_{2}=1$ and $a_{1},\frac{1}{3}c_{D}\geq 2$.From now on, we assume that $2(a_{1}+1)<k\leq\frac{8}{3}(a_{1}+1)$. Note that $D=3$ and $c_{2}\geq 2$ follow by Lemma 4.3. Since there is a quadrangle by Lemma 2.10 (1), there are two vertices $p$ and $q$ at distance two such that vertices $p,q,r_{1},r_{2}$ induce a quadrangle for some $r_{1},r_{2}\in\Gamma_{1}(p,q)$. As $D=3$, there exists $r_{3}\in\Gamma_{3}(p)\cap\Gamma_{1}(q)$. This implies that the set $\{q,r_{1},r_{2},r_{3}\}$ induces a $3$-claw and thus $\theta_{3}\in\{-3,-4,-5\}$ by Eq. (12) and Lemma 2.7 (1). If $\psi_{1}=1$ then $k=(-\theta_{3})(a_{1}+1)$, which is impossible as $k\leq\frac{8}{3}(a_{1}+1)$ and $\theta_{3}\leq-3$. Thus $\psi_{1}\geq 2$. If $\theta_{3}=-3$, then $\Gamma$ is the Johnson graph $J(n,3)$ with $8\leq n\leq 19$ by [4, Theorem 4.3]. By Eq. (12) with $\theta_{3}\in\{-4,-5\}$, there are at least four Delsarte cliques $C_{(i)}~{}(1\leq i\leq 4)$ containing a fixed vertex $x$. If $|C_{(i)}|-1=\frac{k}{-\theta_{3}}\geq 3(\psi_{1}-1)+1$, then there exist vertices $y_{1}\in C_{(1)}$ and $y_{i}\in C_{(i)}\setminus(\cup_{j=1}^{i-1}\Gamma_{1}(y_{j}))~{}(2\leq i\leq 4)$ such that the set $\{x,y_{i}\mid 1\leq i\leq 4\}$ induces a $4$-claw, which is impossible and hence

[TABLE]

It follows by $a_{1}=(\frac{k}{-\theta_{3}}-1)+(-\theta_{3}-1)(\psi_{1}-1)$ with Eq. (13) that if $\theta_{3}=-4$, then

[TABLE]

This is impossible as $k>2(a_{1}+1)$. Hence $\theta_{3}=-5$ and thus $2\leq\psi_{1}\leq 4$ and $\psi_{1}\geq\frac{1}{15}k+1$ follow by Lemma 4.1 (4) and Eq. (13). Using Eq. (13), $\psi_{1}\in\{2,3,4\}$ and $2\psi_{1}\leq 1+\frac{k}{5}$ (see [6, Lemma 5.2 (ii)]) we find $(\psi_{1},k)\in\{(2,15),(3,25),(3,30),(4,35),(4,40),(4,45)\}.$ By the feasible conditions $k_{2},k_{3},b_{1},b_{2},c_{2},c_{3}\in\mathbb{N}$, Eq. (5), Lemma 4.1 and $2\leq\psi_{1}<\psi_{2}$ (see [6, Theorem 5.5]), we find that the feasible intersection arrays for $\Gamma$ are $\{15,8,1;1,8,15\}$, $\{25,12,1;1,12,25\}$ and $\{35,16,1;1,16,35\}$. Since graphs with these intersection arrays are Taylor graphs, it is impossible by Theorem 3.1. This completes the proof.

Remark 4.6

*It seems to be difficult to classify distance-regular graphs with $\theta_{D}=-3$ satisfying $k=3(a_{1}+1)$ and $c_{2}=1$ (see [4, 38]). It follows by [4], [7] and [38] that any distance-regular graph in Theorem 4.5 (9) (i.e., a distance-regular graph with $\theta_{D}=-3$ satisfying $k=3(a_{1}+1)$, $c_{2}=1$ and $a_{1},\frac{1}{3}c_{D}\geq 2$) is one of the following:

(1) A distance-regular graph satisfying $D={\texttt{h}}+2\geq 4$ and*

[TABLE]

(2) A distance-regular graph satisfying $D={\texttt{h}}+2\geq 3$ and

[TABLE]

(3) A halved graph of a distance-biregular graph with vertices of valency $3$ and

[TABLE]

It is unknown whether there exists a constant $Y$ such that any distance-regular graph with $k=3(a_{1}+1)$ and $c_{2}=1$ has diameter $D\leq Y$.

Question 4.7

Is it true that any geometric distance-regular graph with smallest eigenvalue $-t$ has a $t$-claw?

5 Non-geometric distance-regular graphs without $4$-claws

In this section, we classify non-geometric distance-regular graphs with diameter $D\geq 3$ and valency $k>3$ but without $4$-claws (see Theorem 5.1).

Theorem 5.1

Let $\Gamma$ be a non-geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$ but without $4$-claws. Then $\Gamma$ is one of the following graphs:

• (1)

the icosahedron with $\iota(\Gamma)=\{5,2,1;1,2,5\}$;

• (2)

the Klein graph with $\iota(\Gamma)=\{7,4,1;1,2,7\}$;

• (3)

the halved $e$-cube with $\iota(\Gamma)=\{{e\choose 2},{e-2\choose 2},1;1,6,15\}$ with $e\in\{6,7\}$;

• (4)

the Taylor graph with $\iota(\Gamma)=\{k,\frac{k-1}{2},1;1,\frac{k-1}{2},k\}$ with $k\in\{13,17\}$;

• (5)

the Gosset graph with $\iota(\Gamma)=\{27,10,1;1,10,27\}$;

or $\Gamma$ has one of the following intersection arrays:

• (6)

$\iota(\Gamma)=\{44,24,1;1,12,44\}$;

• (7)

$\iota(\Gamma)=\{64,34,1;1,17,64\}$;

• (8)

$\iota(\Gamma)=\{104,54,1;1,27,104\}$.

To prove Theorem 5.1, we first show in Subsection 5.1 that the valency $k$ of $\Gamma$ is bounded above by some constant (see Proposition 5.3). In Subsection 5.2, with the aid of computer we find all feasible intersection arrays for distance-regular graphs with $D\geq 3$ and $3$-claws but without $4$-claws whose valencies satisfy the bounds in Subsection 5.1. Finally, in Subsection 5.3, we show that for some of the intersection arrays we found in Subsection 5.2, there are no corresponding distance-regular graphs without $4$-claws. The proof of Theorem 5.1 is given in Subsection 5.3.

We first need the following lemma.

Lemma 5.2

If $\Gamma$ is a non-geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$, and $3$-claws but without $4$-claws, then $c_{2}\geq 2$ and $k\leq\frac{8}{3}(a_{1}+1)$ hold. Moreover, if $k\leq 2(a_{1}+1)$, then $\Gamma$ is one of the following graphs:

• (1)

the halved $e$-cube with $\iota(\Gamma)=\{{e\choose 2},{e-2\choose 2},1;1,6,15\}$ with $e\in\{6,7\}$;

• (2)

the Taylor graph with $\iota(\Gamma)=\{k,\frac{k-1}{2},1;1,\frac{k-1}{2},k\}$ with $k\in\{13,17\}$;

• (3)

the Gosset graph with $\iota(\Gamma)=\{27,10,1;1,10,27\}$.

*Proof: *If $c_{2}=1$, then $k=3(a_{1}+1)$ and $\Gamma$ is geometric by [4, Theorem 3.2]. As $\Gamma$ is non-geometric and $k>3$, it follows from Proposition 2.11 that $k\leq\frac{8}{3}(a_{1}+1)$ holds. Suppose that $k\leq 2(a_{1}+1)$ holds. By Proposition 2.1, the graph $\Gamma$ is the halved $7$-cube or a Taylor graph. In the latter case, the result follows from Theorem 3.1.

For the rest of this section, let $\Gamma$ be a non-geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$, and $3$-claws but without $4$-claws. By Lemma 5.2, it follows that $k\leq\frac{8}{3}(a_{1}+1)$, and, if $k\leq 2(a_{1}+1)$, then $k\leq 27$. Thus, in what follows, we assume that the valency $k$ satisfies

[TABLE]

Further, let $\theta_{0}=k$ $>$ $\theta_{1}$ $>$ $\ldots$ $>$ $\theta_{D}$ be the $D+1$ distinct eigenvalues of $\Gamma$ with multiplicities $m_{0}=1$, $m_{1}$, $\ldots$, $m_{D}$, respectively. Lemma 2.10 (3) shows that $D\leq 4$. By Lemma 2.7 (1) (with $t=3$ and $2<\delta\leq\frac{8}{3}$), $\theta_{D}>-6$ holds.

5.1 Valency bounds

It is known that for given integer $t\geq 2$, there are only finitely many non-geometric distance-regular graphs with both valency and diameter at least $3$ and with smallest eigenvalue at least $-t$, see [19, Theorem 9.10]. However, in order to perform a computer search of feasible intersection arrays, we need to improve the valency bounds.

In this subsection we obtain a better bound for valency $k$ of a non-geometric distance-regular graph $\Gamma$ with diameter $D\geq 3$ but without $4$-claws.

To obtain a valency bound, we consider the following two cases: $\theta_{1}\geq a_{1}$ and $\theta_{1}<a_{1}$. In the latter case, we also consider several subcases depending on ($m_{1}\geq k$ or $m_{1}<k$) and ($\theta_{D}\in\mathbb{Z}$ or $\theta_{D}\notin\mathbb{Z}$).

The main result of this subsection is the following proposition.

Proposition 5.3

*Let $\Gamma$ be a non-geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$, and $3$-claws but without $4$-claws. Then one of the following holds. *

1. (1)

$\theta_{1}\geq a_{1}$* holds, $D\in\{3,4\}$, and:*

1. (i)

if $\theta_{1}\in\mathbb{Z}$, then $k\leq 476$ and $m_{1}\geq k$, 2. (ii)

if $\theta_{1}\notin\mathbb{Z}$, then $k\leq 952$ and $m_{1}\geq\frac{k}{2}$. 2. (2)

$\theta_{1}<a_{1}$* holds, $D=3$, and:*

1. (i)

if $m_{1}\geq k$, then $k\leq 790$, 2. (ii)

if $m_{1}<k$ and $\theta_{1}\notin\mathbb{Z}$, then $k\leq 36$, 3. (iii)

if $m_{1}<k$, $\theta_{1}\in\mathbb{Z}$ and $\theta_{3}\notin\mathbb{Z}$, then $k\leq 530$, 4. (iv)

if $m_{1}<k$, $\theta_{1}\in\mathbb{Z}$ and $\theta_{3}\in\mathbb{Z}$, then $k\leq 833$.

To prove Proposition 5.3, we need several lemmas.

Lemma 5.4

If $\theta_{1}\in\mathbb{Z}$ and $\theta_{1}\geq a_{1}$, then $m_{1}\geq k$.

*Proof: *If $m_{1}<k$ and $\theta_{1}$ is integral, then it follows by [16, Theorem 4.4.4] that $\theta_{1}+1$ divides $b_{1}$, i.e., $b_{1}=\ell(\theta_{1}+1)$ holds for some integer $\ell\geq 1$.

By Eq. (14), $b_{1}\leq\frac{5}{3}(a_{1}+1)$. As $\ell(a_{1}+1)\leq\ell(\theta_{1}+1)=b_{1}\leq\frac{5}{3}(a_{1}+1)$, we find $\ell=1$ and thus $\theta_{1}=b_{1}-1$. As $\Gamma$ is non-geometric and it satisfies $D\geq 3$, $\theta_{1}\geq a_{1}$ with $m_{1}<k$, and $c_{2}\geq 2$ (see Lemma 5.2), it follows by [16, Theorem 4.4.11] that there are no such graphs. Thus $m_{1}\geq k$, and the lemma is proved.

Lemma 5.5

If $\theta_{1}\geq a_{1}$ and $m_{1}\geq\frac{k}{N}$ both hold for some constant $N\geq 1$, then

[TABLE]

*Proof: *Suppose that $\theta_{1}\geq a_{1}$ and $m_{1}\geq\frac{k}{N}$ hold for some constant $N\geq 1$. It follows by Eq. (4), $m_{1}\geq\frac{k}{N}$ and $\theta_{1}\geq a_{1}$ that

[TABLE]

and hence

[TABLE]

We also have that:

• •

$b_{1}=k-(a_{1}+1)=(\delta-1)(a_{1}+1)$ by Eq. (14),

• •

$b_{3}\leq b_{2}\leq(a_{1}+1)$ by Lemma 2.10 (2),

• •

$c_{2}\geq\frac{a_{1}-6}{6}$ holds by Lemma 4.2 as $\Gamma$ is not geometric,

• •

$c_{3}\geq b_{3}+1-b_{1}+2(a_{1}+2)\geq 1-b_{1}+2(a_{1}+2)>-k+3(a_{1}+1)=(3-\delta)(a_{1}+1)$ by Eq. (14) and Lemmas 2.2 and 2.10 (2),

• •

if $D=4$, then $c_{4}\geq 1-b_{1}+3(a_{1}+2)>-k+4(a_{1}+1)=(4-\delta)(a_{1}+1)$ by Eq. (14) and Lemmas 2.2 and 2.10 (2).

Suppose $a_{1}\geq 7$ so that $a_{1}-6>0$. Then we find by Eq. (3) that

[TABLE]

and hence

[TABLE]

Note that the function $\frac{(\delta^{2}-8\delta+17)(\delta-1)}{(3-\delta)(4-\delta)}$ is strictly increasing on $2<\delta\leq\frac{8}{3}$, therefore

[TABLE]

Suppose $a_{1}\geq 146$. Then $\frac{(a_{1}+1)}{(a_{1}-6)}\leq\frac{21}{20}$ and thus $v<67k$. It follows by Eq. (15) that $a_{1}^{2}<(67N-1)k\leq\frac{8(67N-1)}{3}(a_{1}+1)$ and thus

[TABLE]

which completes the proof.

The following technical lemma will be used later (see Lemma 5.7).

Lemma 5.6

If $\theta_{1}<a_{1}$ and $m_{1}\geq\frac{k}{N}$ for some constant $N\geq 1$, then $D=3$ and

[TABLE]

*Proof: *Suppose that $\theta_{1}<a_{1}$ and $m_{1}\geq\frac{1}{N}k$ hold for some constant $N\geq 1$. By Lemma 2.5, $D=3$ holds. Using Eq. (5) we have

[TABLE]

Note that $m_{2}\geq k$ by [16, Theorem 4.4.4] and thus $m_{3}=v-1-m_{1}-m_{2}\leq v-1-\frac{N+1}{N}k$.

By Lemma 2.10 (4), $\theta_{2}\geq-1$ holds and hence $(\theta_{2}+1)m_{2}\geq 0$. It follows by Eq. (16) that

[TABLE]

Now the result follows as $m_{1}(\theta_{1}+1)\geq\frac{1}{N}k(\theta_{1}+1)>\frac{kb_{1}}{-N(\theta_{3}+1)}$ holds by Theorem 2.3.

Lemma 5.7

If $\theta_{1}<a_{1}$ and $m_{1}\geq k$ both hold, then $k\leq 790$.

*Proof: *By Eq. (14) and Lemmas 2.5, 2.10 (2) and 4.2, we have that

[TABLE]

Claim 5.8

The valency $k$ satisfies

[TABLE]

*Proof of Claim 5.8: * If $a_{1}<\max\{42\delta-35,49\}$, then $k<\max\left\{(42\delta-34)\delta,50\delta\right\}$. Now we assume $a_{1}\geq\max\{42\delta-35,49\}$. We first show $v<(6\delta+3)k$. By Eq. (17), we find

[TABLE]

and thus $v=1+k+k_{2}+k_{3}<(6\delta+3)k$.

As $\theta_{3}>-\frac{3\delta}{2\delta-3}$ holds by Lemma 2.7, we find $\theta_{1}+1>\frac{(2\delta-3)(k-1-a_{1})}{\delta+3}$ by Theorem 2.3. It follows by Lemma 5.6 that

[TABLE]

and thus $k<(a_{1}+1)+\frac{(18\delta^{2}+5\delta-3)(\delta+3)}{(2\delta-3)^{2}}$. As $\delta(a_{1}+1)=k$, we find $a_{1}+1<\frac{(18\delta^{2}+5\delta-3)(\delta+3)}{(\delta-1)(2\delta-3)^{2}}$ and hence $k=\delta(a_{1}+1)<\frac{\delta(18\delta^{2}+5\delta-3)(\delta+3)}{(\delta-1)(2\delta-3)^{2}}$. Now the claim follows.

Evaluating the valency bound from Claim 5.8 for $2<\delta\leq\frac{8}{3}$ shows that $k\leq 790$ holds, and this completes the proof.

Remark 5.9

Lemma 2.6 together with Lemma 2.7 show that the multiplicity $m_{1}$ can be bounded below by $m_{1}\geq\frac{1}{N}k$ with an appropriate constant $N>1$. Therefore the valency $k$ can be bounded above in the same manner as in the proof of Lemma 5.7. However, in the following several lemmas, we will obtain more accurate bounds for $k$ for the case when $\theta_{1}<a_{1}$ and $m_{1}<k$.

Lemma 5.10

Suppose that $\theta_{1}<a_{1}$, $\theta_{1}\not\in\mathbb{Z}$, and $m_{1}<k$. Then $k\leq 36$.

*Proof: *It follows from Lemma 2.5 and [16, Theorem 4.4.4] that $D=3$ and $m_{1}=m_{3}<k\leq m_{2}$, and thus, by Lemma 2.10 (4), $\theta_{2}$ is integral satisfying $\theta_{2}\geq-1$.

We first show that if $\theta_{2}=-1$, then $k\leq 36$. If $c_{2}=1$, then $k\leq 36$ holds by [8, Theorem 4.2]. Let $c_{2}\geq 2$. It follows by Eq. (14) and [8, Theorem 4.2 (iv)] that

[TABLE]

where ${\alpha}:=a_{1}-c_{2}$. It follows from Eq. (18) that $0\leq\alpha\leq 10$, and thus $k\leq 36$ holds, if $\theta_{2}=-1$.

Now we assume that $\theta_{2}\geq 0$. We will show that $k\leq 33$. If follows from Eq. (5), $m_{2}\geq k>m_{1}=m_{3}$ and $\theta_{2}\geq 0$ that

[TABLE]

and hence $\theta_{1}+\theta_{3}\leq\frac{k}{m_{1}}(-1-\theta_{2})<-1-\theta_{2}$ and $\theta_{1}<-\theta_{3}-1-\theta_{2}\leq 5$. Thus, we have that $\theta_{1}<5$ and $|\theta_{3}|\leq 6$.

If $k\geq 31>\frac{5^{2}+6^{2}}{2}>\frac{\theta_{1}^{2}+\theta_{3}^{2}}{2}$, then it follows by $tr(A^{0})=v$, $tr(A^{2}){=vk}$ and $m_{1}\geq\frac{k}{2}$ (see Lemma 2.6 (2)) that

[TABLE]

Hence $2k-6\leq\theta_{1}^{2}+\theta_{3}^{2}<25+36$. This shows that $k\leq 33$ if $\theta_{2}\geq 0$. This completes the proof.

Lemma 5.11

If $\theta_{1}\in\mathbb{Z}$ and $m_{1}<k$, then $D=3$ and $b_{1}=\ell(\theta_{1}+1)$ hold for some $\ell\in\{2,3,4\}$. Moreover,

[TABLE]

*Proof: *Suppose that $\theta_{1}$ is integral with multiplicity $m_{1}<k$. By Lemma 5.4, we have $\theta_{1}<a_{1}$, and, by Lemma 2.5, we have $D=3$.

By [16, Theorem 4.4.4], $b_{1}=\ell(\theta_{1}+1)$ holds for some integer $\ell\geq 1$, and $-1-\frac{b_{1}}{\theta_{1}+1}=-1-\ell$ is an eigenvalue of the local graph $\langle\Gamma_{1}(x)\rangle$ for a vertex $x\in V(\Gamma)$. As $k>2(a_{1}+1)$ holds by Eq. (14), it follows by Lemma 2.7 that $\theta_{3}>-6$ and thus $\ell\in\{1,2,3,4\}$ as $-1-\frac{b_{1}}{\theta_{1}+1}\geq\theta_{3}$ follows by the interlacing result (see [16, Corollary 3.3.2]). Note that if $\ell=1$, then by [16, Theorem 4.4.11] $\Gamma$ is the halved $e$-cube with $e\in\{6,7\}$, or the Gosset graph, i.e., $\Gamma$ satisfies $k\leq 2(a_{1}+1)$, a contradiction.

Suppose $\ell\in\{2,3,4\}$. By Eq. (14), $b_{1}=(\delta-1)(a_{1}+1)$ holds. By Lemma 2.4 (1), we have $b_{1}=\ell(\theta_{1}+1)>\ell\theta_{1}>\ell(a_{1}+1-c_{2})$ and thus

[TABLE]

(Note here that $\ell-\delta+1>0$ holds from $b_{1}=(\delta-1)(a_{1}+1)=\ell(\theta_{1}+1)<\ell(a_{1}+1)$.)

By Lemma 2.4 (2) with $\theta_{1}<a_{1}$, we have $a_{3}\leq\theta_{1}<a_{1}$, and thus $c_{3}=k-a_{3}\geq k-\theta_{1}>k-(a_{1}+1)=b_{1}=(\delta-1)(a_{1}+1)$. By Lemma 2.10, we have

[TABLE]

Hence it follows by $c_{3}>b_{1}$ and Eq. (19) that

[TABLE]

Now the result follows as $v=1+k+k_{2}+k_{3}$. In particular, using Eq. (19) and $a_{3}\leq\theta_{1}<a_{1}$,

[TABLE]

This shows $v=1+k+k_{2}+k_{3}<1+k+2k_{2}<1+k+\frac{2(\delta-1)\ell}{(\ell-\delta+1)}k$.

Lemma 5.12

Suppose that $\theta_{1}<a_{1}$, $\theta_{1}\in\mathbb{Z}$, and $m_{1}<k$. If the smallest eigenvalue of $\Gamma$ is not integral, then $k\leq 530$.

*Proof: *By Lemma 2.5, we have $D=3$. Suppose that $\theta_{3}\notin\mathbb{Z}$. Then the eigenvalues $\theta_{2}$ and $\theta_{3}$ are conjugate algebraic integers with $m_{2}=m_{3}$ as $\theta_{1}\in\mathbb{Z}$. By Eq. (5), we have

[TABLE]

This shows

[TABLE]

By Lemma 5.11, $b_{1}=\ell(\theta_{1}+1)=(\delta-1)(a_{1}+1)$ holds for some $\ell\in\{2,3,4\}$. In particular,

[TABLE]

holds for some $\frac{\ell}{a_{1}+1}\leq\epsilon\leq\delta-1$. If $a_{1}+1<200$, then $k\leq 530$ as $k\leq\frac{8}{3}(a_{1}+1)$.

In the rest of the proof, we assume $a_{1}+1\geq 200$ and $\epsilon=0.02$. It follows by Lemma 2.10 (4) that $0<-(\theta_{2}+\theta_{3})\leq 1-\theta_{3}$. We find by Eqs. (20) and (21) that

[TABLE]

By Lemma 5.11 and Eq. (22),

[TABLE]

follows and thus

[TABLE]

We now put $M(\delta,\theta_{3},\ell):=\frac{\ell(1-\theta_{3})}{2}\times\frac{A(\delta,\theta_{3},\ell)}{B(\delta,\ell)}$, where

[TABLE]

Then

[TABLE]

To determine an upper bound in Eq. (24), we first show the following claim.

Claim 5.13

*Let $\tau$ and $\gamma$ be some real numbers.

(1) If $\tau\leq\theta_{3}\leq-2$ holds, then $M(\delta,\theta_{3},\ell)\leq M(\delta,\tau,\ell)$.

(2) If $2<\delta\leq\gamma\leq\frac{8}{3}$ holds, then $M(\delta,\theta_{3},\ell)\leq M(\gamma,\theta_{3},\ell)$.*

*Proof of Claim 5.13: * (1) For given $\delta$ and $\ell$, function $A(\delta,\theta_{3},\ell)(1-\theta_{3})$ of $\theta_{3}$ is decreasing on $\theta_{3}\leq-2$ as

[TABLE]

Thus $M(\delta,\theta_{3},\ell)$ is also decreasing on $\theta_{3}\leq-2$, and this shows part (1) as $\tau\leq\theta_{3}\leq-2$.

(2) As $M(\delta,\theta_{3},\ell)=\frac{\ell(1-\theta_{3})}{2}\times\frac{A(\delta,\theta_{3},\ell)}{B(\delta,\ell)}$, it is enough to show that $\frac{\partial}{\partial\delta}\left(\frac{A(\delta,\theta_{3},\ell)}{B(\delta,\ell)}\right)>0$ holds for all $2<\delta\leq\frac{8}{3}$. Note that $2<\delta<3\leq\ell+1$. By Eq. (23), we have

[TABLE]

If $\ell-2\delta+2.02<0$ then $\frac{\partial}{\partial\delta}\left(\frac{A(\delta,\theta_{3},\ell)}{B(\delta,\ell)}\right)>0$ holds by Eq. (5.1). Now we suppose that $\ell-2\delta+2.02\geq 0$ holds. Then for each $\ell\in\{2,3,4\}$, $2<\delta\leq\min\left\{\frac{\ell+2.02}{2},\frac{8}{3}\right\}$ holds and thus

[TABLE]

This yields that the right hand side in Eq. (5.1) is positive. This proves (2) and hence the claim follows.

To complete the proof of this lemma, we consider the following two cases.

If $\frac{7}{3}<\delta\leq\frac{8}{3}$, then it follows by Lemma 2.7 (1) that there exists a clique $C$ satisfying $|C|>\frac{5k}{21}+1$ and thus $\theta_{3}>-\frac{21}{5}$. By Eq. (24) and Claim 5.13, we find $m_{1}\leq 124$ as

[TABLE]

It follows by Lemma 2.6 (1) that $\frac{5k}{21}+1\leq m_{1}\leq 124$ and thus $k\leq 516$.

Now we suppose that $2<\delta\leq\frac{7}{3}$. By Lemma 2.7 (1), there exists a clique $C$ satisfying $|C|>\frac{k}{6}+1$ and thus $\theta_{3}>-6$. Now we find $m_{1}\leq 89$ as

[TABLE]

holds by Eq. (24) and Claim 5.13. Hence it follows by Lemma 2.6 (1) that $\frac{k}{6}+1\leq m_{1}\leq 89$ and thus $k\leq 528$.

This shows that if $a_{1}+1\geq 200$, then $k\leq 528$. Recall that $k\leq 530$, if $a_{1}+1<200$. This completes the proof.

Lemma 5.14

Suppose that $\theta_{1}<a_{1}$, $\theta_{1}\in\mathbb{Z}$ and $m_{1}<k$. If the smallest eigenvalue of $\Gamma$ is integral, then $k\leq 833$.

*Proof: *By Eq. (14), Lemmas 5.2 and 5.11, we have that $k=\delta(a_{1}+1)$ for some number $2<\delta\leq\frac{8}{3}$, $b_{1}=\ell(\theta_{1}+1)$ for some $\ell\in\{2,3,4\}$, and $D=3$.

Using Eq. (5), we have $v=tr(A^{0})+tr(A)=k+1+(\theta_{1}+1)m_{1}+(\theta_{2}+1)m_{2}+(\theta_{3}+1)m_{3}.$ Note that $m_{2}\geq k$ by [16, Theorem 4.4.4] and thus $m_{3}=v-1-m_{1}-m_{2}\leq v-2-k$. As $\theta_{2}\geq-1$ holds by Lemma 2.10 (4), we obtain that

[TABLE]

By Lemma 5.11, $b_{1}=\ell(\theta_{1}+1)$ and $k=\frac{b_{1}\delta}{\delta-1}$, it follows that

[TABLE]

To complete the proof of the lemma, we consider the following three cases.

If $2<\delta\leq\frac{7}{3}$, then it follows by Lemma 2.7 (1) that there exists a clique $C$ satisfying $|C|>\frac{k}{6}+1$ and thus $\theta_{3}\geq-5>-6$. Hence it follows by Lemma 2.6 (1) and Eq. (26) with $(\delta,\theta_{3},\ell)=(\frac{7}{3},-5,2)$ that $\frac{k}{6}+1<m_{1}\leq 140$. This shows $k<834$.

If $\frac{7}{3}<\delta\leq\frac{5}{2}$, then it follows by Lemma 2.7 (1) that there exists a clique $C$ satisfying $|C|>\frac{5k}{21}+1$ and thus $\theta_{3}\geq-4>-\frac{21}{5}$. Hence it follows by Lemma 2.6 (1) and Eq. (26) with $(\delta,\theta_{3},\ell)=(\frac{5}{2},-4,2)$ that $\frac{5k}{21}+1<m_{1}\leq 160$. This shows $k<668$.

If $\frac{5}{2}<\delta\leq\frac{8}{3}$, then it follows by Lemma 2.7 (1) that there exists a clique $C$ satisfying $|C|>\frac{4k}{15}+1$ and thus $\theta_{3}\geq-3>-\frac{15}{4}$. Hence it follows by Lemma 2.6 (1) and Eq. (26) with $(\delta,\theta_{3},\ell)=(\frac{8}{3},-3,2)$ that $\frac{4k}{15}+1<m_{1}\leq 192$. This shows $k<717$.

This completes the proof.

*Proof of Proposition 5.3: * By Lemma 2.10 (3), $D\in\{3,4\}$ holds.

(1) Suppose that $\theta_{1}\geq a_{1}$ holds. If $\theta_{1}\in\mathbb{Z}$, then $m_{1}\geq k$ holds by Lemma 5.4, and thus $k\leq 476$ by Lemma 5.5. If $\theta_{1}\notin\mathbb{Z}$, then $m_{1}\geq k/2$ holds by Lemma 2.6 (2), and thus $k\leq 952$ by Lemma 5.5. This shows (1).

(2) Suppose that $\theta_{1}<a_{1}$. Then $D=3$ holds by Lemma 2.5. The statements (i)–(iv) follow by Lemmas 5.7, 5.10, 5.12, 5.14, respectively. This completes the proof.

5.2 Computational results

In this subsection, with the aid of computer, we find all feasible intersection arrays for distance-regular graphs with $D\geq 3$ and $3$-claws but without $4$-claws whose valency satisfies the bounds in Proposition 5.3.

We call an intersection array $\{b_{0},\ldots,b_{D-1};1,c_{2},\ldots,c_{D}\}$ feasible, if it satisfies the following conditions:

• $(F1)$

$k_{i}\in\mathbb{N}~{}~{}(i=2,\ldots,D)$;

• $(F2)$

$ka_{1}\equiv\ldots\equiv k_{D}a_{D}\equiv 0~{}({\rm mod}~{}2)$ (the handshake lemma);

• $(F3)$

$vk_{i}a_{i}\equiv 0~{}({\rm mod}~{}6)$ (see [16, Lemma 4.3.1]), where $v=|V(\Gamma)|=\sum_{i=0}^{D}k_{i}$;

• $(F4)$

all multiplicities $m_{i}$ calculated by Eq. (4) are positive integers;

• $(F5)$

all intersection numbers $p_{ij}^{k}$ are non-negative integers.

Suppose that $\Gamma$ is a non-geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$, and $3$-claws but without $4$-claws.

For each of the following cases given by Proposition 5.3:

1. $(C1)$

$\theta_{1}\geq a_{1}$, $\theta_{1}\in\mathbb{Z}$, $m_{1}\geq k$, $D\in\{3,4\}$, and $k\leq 476$; 2. $(C2)$

$\theta_{1}\geq a_{1}$, $\theta_{1}\notin\mathbb{Z}$, $m_{1}\geq\frac{k}{2}$, $D\in\{3,4\}$ and $k\leq 952$; 3. $(C3)$

$\theta_{1}<a_{1}$, $m_{1}\geq k$, $D=3$, and $k\leq 790$; 4. $(C4)$

$\theta_{1}<a_{1}$, $m_{1}<k$, $\theta_{1}\notin\mathbb{Z}$, $D=3$, and $k\leq 36$; 5. $(C5)$

$\theta_{1}<a_{1}$, $m_{1}<k$, $\theta_{1}\in\mathbb{Z}$, $\theta_{3}\notin\mathbb{Z}$, $D=3$, and $k\leq 530$; 6. $(C6)$

$\theta_{1}<a_{1}$, $m_{1}<k$, $\theta_{1}\in\mathbb{Z}$, $\theta_{3}\in\mathbb{Z}$, $D=3$, and $k\leq 833$,

we check by computer all feasible intersection arrays additionally satisfying:

• •

$2(a_{1}+1)<k\leq\frac{8}{3}(a_{1}+1)$ (see Lemma 5.2),

• •

$\theta_{D}>-6$ (by Lemma 2.7 with $t=3$ and $\delta=2$),

• •

$c_{2}\geq\max\left\{2,\frac{a_{1}-6}{6}\right\}$ (see Lemma 2.10 (1) and Lemma 4.2), and

• •

$b_{2}\leq a_{3}+1$ (by Lemma 2.10 (2)).

For a given intersection array, we successively check the properties $(Fi)$, $i=1,2,\ldots,5$. If it passes $(F1)$–$(F3)$, we then check $(F4)$ by calculating the eigenvalues and their multiplicities. For both cases, $D=3$ and $D=4$, we apply a root-finding algorithm known as Brent’s method [15] for the characteristic polynomial of the matrix $L_{1}$. For $D=3$, if we find one eigenvalue then we apply the observation in [16, p. 130] in order to calculate two remaining non-trivial eigenvalues. For $D=4$, we look for two roots in the intervals $(-6,-2)$ and $[a_{1},b_{1}-1)$, respectively (as $\theta_{1}\leq b_{1}-1$ by [16, Proposition 4.4.9(i)] and $\theta_{1}\geq a_{1}$ by the assumption). If two such roots are found, then we proceed as in the case $D=3$. Finally, if all the multiplicities $m_{i}$, $i=1,\ldots,D$ are integral and positive, then we check whether $(F5)$ holds.

We can now summarize the computational results.

Proposition 5.15

The following holds.

1. (1)

The only feasible intersection arrays satisfying $(C1)$ are:

[TABLE] 2. (2)

The only feasible intersection array satisfying $(C2)$ is:

[TABLE] 3. (3)

The only feasible intersection arrays satisfying $(C3)$ are:

[TABLE] 4. (4)

There are no feasible intersection arrays satisfying $(C4)$. 5. (5)

There are no feasible intersection arrays satisfying $(C5)$. 6. (6)

The only feasible intersection arrays satisfying $(C6)$ are:

[TABLE]

[TABLE]

[TABLE]

5.3 Non-existence of some distance-regular graphs without $4$-claws

In this subsection, we show that for some of the intersection arrays found in Subsection 5.2 (see Proposition 5.15) there are no corresponding distance-regular graphs without $4$-claws.

Lemma 5.16

There are no $4$-claw-free distance-regular graphs with intersection arrays satisfying $(C1)$:

[TABLE]

*Proof: *The intersection array $\{10,6,1;1,3,10\}$ is ruled out by [16, Proposition 1.10.5], and the last three arrays are ruled out by Lemma 2.10 (1) and Lemma 2.2. Indeed, they satisfy $k>2(a_{1}+1)$ and $k<3a_{1}-3c_{2}+7$, and hence any distance-regular graph with one of these three intersection arrays must be a Terwilliger graph by Lemma 2.2. However, this is impossible by Lemma 2.10 (1).

Lemma 5.17

There are no $4$-claw-free distance-regular graphs with the following intersection arrays satisfying $(C6)$:

[TABLE]

[TABLE]

*Proof: *The intersection arrays $\{125,78,1;1,26,125\}$, $\{155,96,1;1,32,155\}$, $\{275,168,1;1,56,275\}$, $\{575,348,1;1,116,575\}$ and $\{715,432,25;1,80,675\}$ satisfy $\theta_{3}=-5$ and $\frac{5}{2}(a_{1}+1)<k\leq\frac{8}{3}(a_{1}+1)$. This is impossible as $\theta_{3}\geq-3$ follows from Lemma 2.7 (1) (see the proof of Lemma 5.14).

A distance-regular graph $\Gamma$ with intersection array $\{15,8,1;1,8,15\}$ has eigenvalues $\theta_{1}=3$ and $\theta_{3}=-5$ with multiplicities $m_{1}=10$ and $m_{3}=6$ by Eq. (4). It follows by [16, Theorem 4.4.4] that each local graph of $\Gamma$ has eigenvalues $-1-\frac{8}{-5+1}=1$ with multiplicity $15-6=9$ and $-1-\frac{8}{3+1}=-3$ with multiplicity $15-10=5$. Hence each local graph is a strongly regular graph with parameters $(15,6,1,3)$, i.e. the unique generalized quadrangle $GQ(2,2)$. Since $GQ(2,2)$ is the complement of the triangular graph $T(6)$, which has a 4-clique, $GQ(2,2)$ contains a 4-coclique and thus $\Gamma$ has a 4-claw. This is impossible.

Remark 5.18

We note that there exists a distance-regular graph with intersection array $\{44,24,1;1,12,44\}$. It was constructed in [26]. One can check that this graph contains $4$-claws, however, it is an open question whether a distance-regular graph with this intersection array must have a $4$-claw or not.

In the rest of this subsection, we study this problem for the intersection arrays satisfying $(C3)$. We note that there exists a distance-regular graph with intersection array $\{13,8,1;1,4,13\}$, given by the Mathon construction, see [16, Proposition 12.5.3]. There also exists a distance-regular graph $\Gamma$ with intersection array $\{16,10,1;1,5,16\}$, which is locally folded 5-cube, see [16, p. 386, Remark (iii)].

Lemma 5.19

Any distance-regular graph with intersection array $\{13,8,1;1,4,13\}$ or $\{16,10,1;1,5,16\}$ contains a $4$-claw.

*Proof: *Suppose that $\Gamma$ is a $4$-claw-free distance-regular graph with intersection array $\{13,8,1;1,4,13\}$ or $\{16,10,1;1,5,16\}$. Note that $\theta_{3}=-\sqrt{k}$ for both possibilities.

The Delsarte bound (see Eq. (1)) for $\Gamma$ shows that a clique in $\Gamma$ contains at most $\lfloor 1+\sqrt{k}\rfloor$ vertices.

Pick a vertex $x\in V(\Gamma)$, and let $y_{1},y_{2}$ be two non-adjacent vertices of $\Gamma_{1}(x)$. As $\Gamma$ is $4$-claw-free, the set \langle\{x\}\cup\Gamma_{1}(x)\setminus\big{(}\{y_{1},y_{2}\}\cup\Gamma_{1}(y_{1})\cup\Gamma_{1}(y_{2})\big{)}\rangle induces a clique of size $1+k-(2(a_{1}+1-\nu)+\nu)=k-2a_{1}-1+\nu$, where $\nu=|\Gamma_{1}(x,y_{1},y_{2})|$. For $(k,a_{1})\in\{(13,4),(16,5)\}$, we have $k-2a_{1}-1+\nu\leq\lfloor 1+\sqrt{k}\rfloor$. Thus $\nu=0$.

Now pick a vertex $y_{3}$ from Y:=\Gamma_{1}(x)\setminus\big{(}\{y_{1},y_{2}\}\cup\Gamma_{1}(y_{1})\cup\Gamma_{1}(y_{2})\big{)}. As $|\Gamma_{1}(x,y_{i},y_{j})|=0$ holds for any $1\leq i<j\leq 3$, we see that $k=|\Gamma_{1}(x)|\geq\sum_{i=1}^{3}|\{y_{i}\}\cup\Gamma_{1}(x,y_{i})|=3(a_{1}+1)$. This is impossible, as $k\leq\frac{8}{3}(a_{1}+1)$, and the lemma follows.

Proposition 5.20

There are no distance-regular graphs with intersection array $\{39,24,1;1,4,39\}$.

*Proof: *Let $\Gamma$ be a distance-regular graph with intersection array $\{39,24,1;1,4,39\}$. Then $\theta_{0}=39$, $\theta_{1}=13$, $\theta_{2}=-1$, $\theta_{3}=-3$, and $m_{0}=1$, $m_{1}=45$, $m_{2}=39$, $m_{3}=195$.

For a vertex $\infty\in V(\Gamma)$, if the local graph $\Delta:=\langle\Gamma_{1}(\infty)\rangle$ contains an $s$-coclique then

[TABLE]

holds by [29, Lemma 2]. This yields that $\Gamma$ is $4$-claw-free, as $4\cdot 15-39>6\cdot 3$.

Let $\{x_{1},x_{2},x_{3}\}$ induce a $3$-coclique in $\Delta$. As $\Gamma$ is $4$-claw-free, we see that

[TABLE]

so that $\sum_{1\leq i<j\leq 3}|\Delta_{1}(x_{i},x_{j})|\geq 6$. This implies that $x_{i}$ lies in a maximal clique of $\Delta$, say $C_{x_{i}}$, with $\gamma_{i}\geq 9$ vertices, $i\in\{1,2,3\}$. The Delsarte bound (see Eq. (1)) for $\Gamma$ shows that $\gamma_{i}\leq 13$.

The number of vertices of $\Delta\setminus\{C_{x_{1}},C_{x_{2}},C_{x_{3}}\}$ is equal to

[TABLE]

so that $\gamma_{1}+\gamma_{2}+\gamma_{3}\geq 33$.

Let ${\cal C}$ be the set of maximal cliques in $\Delta$ with at least $9$ vertices. Then $C_{x_{1}},C_{x_{2}},C_{x_{3}}\in{\cal C}$, and for every vertex $x\in\Delta$, there exists a clique $C_{x}\in{\cal C}$, $|C_{x}|\leq 13$. As $c_{2}=4$, we have $|C\cap C^{\prime}|\leq 3$ for all distinct $C,C^{\prime}\in{\cal C}$. Therefore ${\cal C}$ satisfies one of the following:

• (a)

${\cal C}=\{C_{1},C_{2},C_{3}\}$, where every $C_{i}\cup\{\infty\}$ is a Delsarte clique in $\Gamma$;

• (b)

${\cal C}=\{C_{1},C_{2},C_{3},C_{4}\}$, where $|C_{1}|=13$, $|C_{2}|=11$, $|C_{3}|=|C_{4}|=9$, and $C_{3}$ and $C_{4}$ intersect in exactly three vertices;

• (c)

${\cal C}=\{C_{1},C_{2},C_{3},C_{4}\}$, where $|C_{1}|=12$, $|C_{2}|=12$, $|C_{3}|=|C_{4}|=9$, and $C_{3}$ and $C_{4}$ intersect in exactly three vertices.

Claim 5.21

Case (b) is impossible.

Proof of Claim 5.21: In this case there exists a vertex $x\in C_{4}\setminus C_{3}$ with 6 neighbours in $C_{1}\cup C_{2}$ ($x$ has valency 14, and has no neighbours in $C_{3}\setminus C_{4}$), but there are at most three of its neighbours in $C_{1}$ and at most three of its neighbours in $C_{2}$. This implies that $x$ has at least four neighbours in $C_{1}\cup\{\infty\}$, which is a Delsarte clique in $\Gamma$, a contradiction. This shows the claim.

Thus, by Claim 5.21 and the connectivity of $\Gamma$, for all vertices $x$ in $\Gamma$, the sets ${\cal C}_{x}$ of maximal cliques in $\langle\Gamma_{1}(x)\rangle$ with at least $9$ vertices satisfy either Case $(a)$ or Case $(c)$ from the above.

In Case (a), $\Gamma$ is geometric, a contradiction.

In Case (c), the number of cliques in $\Gamma$ with exactly $13$ vertices is equal to $2v/13=2\cdot 280/13$ and hence is non-integer, a contradiction, and the proposition follows.

We close this section by proving Theorem 5.1.

*Proof of Theorem 5.1: * Let $\Gamma$ be a non-geometric distance-regular graph with diameter $D\geq 3$, valency $k>3$ but without $4$-claws. If $\Gamma$ is $3$-claw-free, then $\Gamma$ is the icosahedron by [11]. Suppose that $\Gamma$ contains a $3$-claw. If $k\leq 2(a_{1}+1)$ holds, then, by Lemma 5.2, $\Gamma$ satisfies (3), (4), or (5) of Theorem 5.1. If $k>2(a_{1}+1)$, then it follows by Proposition 5.15, Lemmas 5.16, 5.17, 5.19, and Proposition 5.20 that $\Gamma$ has the intersection array $\{7,4,1;1,2,7\}$, i.e., $\Gamma$ is the Klein graph, or one of the intersection arrays given in (6)–(8) of Theorem 5.1. This completes the proof.

6 Proof of Theorem 1.1

*Proof of Theorem 1.1: * This is straightforward by Proposition 2.11, Theorems 3.1, Theorem 4.5, and Theorem 5.1.

Acknowledgements

Sejeong Bang was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2011-0013985).

The research of Alexander Gavrilyuk was funded by Chinese Academy of Sciences President’s International Fellowship Initiative (Grant No. 2016PE040). His work (e.g., Proposition 5.15) was also partially supported by the Russian Science Foundation (grant 14-11-00061).

Jack Koolen was partially supported by the National Natural Science Foundation of China (No. 11471009 and No. 11671376).