A spectral characterization of the $s$-clique extension of the triangular graphs

TL;DR

This paper proves that for large enough n, co-edge-regular graphs cospectral with the s-clique extension of triangular graphs are uniquely characterized as such extensions, revealing a spectral characterization of these graph classes.

Contribution

It establishes a spectral characterization for the s-clique extension of triangular graphs, showing uniqueness for large n among co-edge-regular graphs.

Findings

Co-edge-regular graphs cospectral with s-clique extensions are uniquely identified as such for large n.

Spectral properties determine the structure of s-clique extensions of triangular graphs.

The result applies for all integers s ≥ 2 and sufficiently large n.

Abstract

A regular graph is co-edge regular if there exists a constant $\mu$ such that any two distinct and non-adjacent vertices have exactly $\mu$ common neighbors. In this paper, we show that for integers $s\ge 2$ and $n$ large enough, any co-edge-regular graph which is cospectral with the $s$-clique extension of the triangular graph $T((n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$.

Equations

$$\{(2sn-3s-1{)}^1,(sn-3s-1{)}^{n-1},(-s-1{)}^{\frac{n^2-3n}{2}},(-1{)}^{\frac{(s-1)n(n-1)}{2}}\},$$

$${\theta }_i\ge {\mu }_i\ge {\theta }_{n-m+i},i=1,2,\dots ,m,$$

$$Q_{i,j}=\frac{1}{V_i}(P^{T}NP{)}_{i,j}.$$

$${\theta }_0^{m_0},{\theta }_1^{m_1},\dots ,{\theta }_t^{m_t},$$

$$\{(s({\theta }_0+1)-1{)}^{m_0},(s({\theta }_1+1)-1{)}^{m_1},\dots ,(s({\theta }_t+1)-1{)}^{m_t},(-1{)}^{(s-1)(m_0+m_1+\cdots +m_t)}\}.$$

$$A^3-(\sum _{i=1}^3{\theta }_i)A^2+(\sum _{1\le i<j\le 3}{\theta }_i{\theta }_j)A-{\theta }_1{\theta }_2{\theta }_{3}I=\frac{{\prod }_{i=1}^3(k-{\theta }_i)}{|V(G)|}J.$$

$$\{{\theta }_0^{m_0},{\theta }_1^{m_1},{\theta }_2^{m_2},{\theta }_3^{m_3}\}=\{(s(2n-3)-1{)}^1,(s(n-3)-1{)}^{n-1},(-s-1{)}^{\frac{n^2-3n}{2}},(-1{)}^{(s-1)\frac{n(n-1)}{2}}\}.$$

$$A^3+(3+4s-sn)A^2+((3-n)s^2+(8-2n)s+3)A+(1-(n-4)s-(n-3)s^2)I=4s^2(2n-3)J.$$

$$A_{xy}^3=\{\begin{array}{ll}2s^2{n}^2-2s^{2}n-6sn-3s^2+9s+2,& \text{if}x=y;\\ 9s^{2}n+2sn-15s^2-8s-3-(3+4s-sn){\lambda }_{xy},& \text{if}x\sim y;\\ 8s^{2}n-12s^2-(3+4s-sn){\mu }_{xy},& \text{if}x\nsim y.\end{array}$$

$$\alpha (X)\le \frac{|X|(-\tau )}{k-\tau }.$$

$$c\le s(n-1).$$

$$\sum _{i=1}^k{d}_i=2\epsilon =2s^2{n}^2-2s^{2}n-6sn-3s^2+9s+2,$$

$$\sum _{i=1}^k(d_i{)}^2=2sn(s^2{n}^2-6sn-6s^2+10s+8)+9s^3+3s^2-24s-4,$$

$$\sum _{i=1}^k(d_i-(sn-2){)}^2=(s-1)s^2(n-3{)}^2.$$

$$(s-1)s^2(n-3{)}^2=\sum _{y\sim \infty }(d_y-(sn-2){)}^2$$

$$\ge \sum _{y\in F}(\frac{1}{4}(sn-2){)}^2$$

$$=\frac{1}{16}f(sn-2{)}^2$$

$$\ge \frac{1}{16}f(sn-s{)}^2.$$

$$f\le 16(s-1),$$

$$e=k-f>sn.$$

$$A_i:=\{y\sim \infty \mid y\sim x_i,y\nsim x_j\text{mbox}forall{x}_j\in X,j\ne i\}\cup \{x_i\}.$$

$$\frac{3\times (\frac{3}{4}s(n-1)+1)-k}{3}\le 4s-1.$$

$$|V(C_2)|\ge k-16(s-1)-s(n-1)>\frac{3}{4}s(n-1),$$

$$s(n-3)+1\le c_i\le s(n-1)$$

$$2sn-19s\le k-1-\ell \le d_y\le k-1=2sn-3s-2,$$

$$d_y\le 2\times (4s-1)+\ell -1\le 2(4s-1)+16(s-1)-1\le 24s.$$

$$(s-1)s^2(n-3{)}^2$$

$$\ge \sum _{y\in L}(d_y-(sn-2){)}^2+\sum _{y\in M}(d_y-(sn-2){)}^2$$

$$\ge \ell ((sn-s)-24s{)}^2+m((2sn-19s)-sn{)}^2$$

$$=\ell s^2(n-25{)}^2+m{s}^2(n-19{)}^2$$

$$\ge (\ell +m)s^2(n-25{)}^2.$$

$$\ell +m\le \frac{(s-1)(n-3{)}^2}{(n-25{)}^2}<s$$

$$\ell +m\le s-1.$$

$$d_y-(sn-2)\le k-1-(sn-2)=sn-3s.$$

Full text

A spectral characterization of the $s$-clique extension of the triangular graphs

Ying-Ying Tan

School of Mathematics $\&$ Physics, Anhui Jianzhu University, Hefei, Anhui, 230201, PR China.

Jack H. Koolen111Corresponding author. Email addresses: koolen@ustc.edu.cn (J.H. Koolen), tansusan1@ahjzu.edu.cn (Y.-Y. Tan), xzj@mail.ustc.edu.cn (Z.-J. Xia).

School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, PR China

Wen-Tsun Wu Key Laboratory of the CAS, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, PR China

Zheng-Jiang Xia

School of Finance, Anhui University of Finance and Economics, Bengbu, Anhui, 233030, PR China.

Abstract: A regular graph is co-edge regular if there exists a constant $\mu$ such that any two distinct and non-adjacent vertices have exactly $\mu$ common neighbors. In this paper, we show that for integers $s\geq 2$,and $n$ large enough, any co-edge-regular graph which is cospectral with the $s$-clique extension of the triangular graph $T(n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$.

Keywords: co-edge-regular graph, $s$-clique extension, triangular graph.

AMS classification: 05C50, 05C75, 05C62

This paper is dedicated to the memory of Prof. Slobadan Simi$\rm\acute{c}$.

1 Introduction

All graphs in this paper are simple and undirected. For definitions related to distance-regular graphs, see[1, 11]. Before we state the main result, we give more definitions.

Let $G$ be a simple connected graph on vertex set $V(G)$, edge set $E(G)$ and adjacency matrix $A$. The eigenvalues of $G$ are the eigenvalues of $A$. Let $\lambda_{0},\lambda_{1},\ldots,\lambda_{t}$ be the distinct eigenvalues of $G$ and $m_{i}$ be the multiplicity of $\lambda_{i}$ ($i=0,1,\ldots,t$). Then the multiset $\{\lambda_{0}^{m_{0}},\lambda_{1}^{m_{1}},\ldots,\lambda_{t}^{m_{t}}\}$ is called the spectrum of $G$. Two graphs are called cospectral if they have the same spectrum. Note that a graph $H$ cospectral with a $k$-regular graph $G$ is also $k$-regular.

Recall that a regular graph is co-edge-regular if there exists a constant $\mu$ such that any two distinct and non-adjacent vertices have exactly $\mu$ common neighbors. Our main result in this paper is as follows:

Theorem 1.1

Let $\Gamma$ be a co-edge-regular graph with spectrum

[TABLE]

where $s\geq 2$ and $n\geq 1$ are integers. If $n\geq 48s$, then $\Gamma$ is the $s$-clique extension of the triangular graph $T(n)$.

This paper is a follow-up paper of Hayat, Koolen and Riaz[4]. They showed a similar result for the square grid graphs. In that paper, they gave the following conjecture.

Conjecture 1.2

[4]** Let $\Gamma$ be a connected co-edge-regular graph with four distinct eigenvalues. Let $t\geq 2$ be an integer and $|V(\Gamma)|=n(\Gamma)$. Then there exists a constant $n_{t}$ such that, if $\theta_{min}(\Gamma)\geq-t$ and $n(\Gamma)\geq n_{t}$ both hold, then $\Gamma$ is the $s$-clique extension of a strongly regular graph for some $2\leq s\leq t-1$.

This conjecture is wrong as the $p\times q$-grids $(p>q\geq 2)$ show. So we would like to modify this conjecture as follows.

Conjecture 1.3

Let $\Gamma$ be a connected co-edge-regular graph with parameters $(n,k,\mu)$ having four distinct eigenvalues. Let $t\geq 2$ be an integer. Then there exists a constant $n_{t}$ such that, if $\theta_{min}(\Gamma)\geq-t$, $n\geq n_{t}$ and $k<n-2-\frac{(t-1)^{2}}{4}$, then either $\Gamma$ is the $s$-clique extension of a strongly regular graph for $2\leq s\leq t-1$ or $\Gamma$ is a $p\times q$-grid with $p>q\geq 2$.

The reason for the valency condition is, that in [12], it was shown that for $\lambda\geq 2$, there exist constants $C(\lambda)$ such that a connected $k$-regular co-edge-regular graph with order $v$ and smallest eigenvalue at least $-\lambda$ satisfies one of the following:

• (i)

$v-k-1\leq\frac{(\lambda-1)^{2}}{4}+1$, or;

• (ii)

Every pair of distinct non-adjacent vertices have at most $C(\lambda)$ common neighbours.

Koolen et al.[8] improved this result by showing that one can take $C(\lambda)=(\lambda-1)\lambda^{2}$ if $k$ is much larger than $\lambda$. This paper is part of the project to show the conjecture for $t=3$.

Another motivation comes from the lecture notes [9]. In these notes, Terwilliger shows that any local graph of a thin $Q$-polynomial distance-regular graph is co-edge-regular and has at most five distinct eigenvalues. So it is interesting to study co-edge-regular graphs with a few distinct eigenvalues.

We mainly follow the method of Hayat et al.[4]. The main difference is that we simplify the method of Hayat et al. when we show that every vertex lies on exactly two lines. This leads to a better bound for which we can show this. This will also improve the bound given in the result of Hayat et al.

2 Preliminaries

2.1 Definitions

For two distinct vertices $x$ and $y$, we write $x\sim y$ (resp. $x\nsim y$) if they are adjacent (resp. nonadjacent) to each other. For a vertex $x$ of $G$, we define $N_{G}(x)=\{y\in V(G)\mid y\sim x\}$, and $N_{G}(x)$ is called the neighborhood of $x$. The graph induced by $N_{G}(x)$ is called the local graph of $G$ with respect to $x$ and is denoted by $G(x)$. We denote the number of common neighbors between two distinct vertices $x$ and $y$ by $\lambda_{x,y}$ (resp. $\mu_{x,y}$) if $x\sim y$ (resp. $x\nsim y$).

A graph is called regular if every vertex has the same valency. A regular graph $G$ with $n$ vertices and valency $k$ is called co-edge-regular with parameters $(n,k,\mu)$ if any two nonadjacent vertices have exactly $\mu=\mu(G)$ common neighbors. In addition, if any two adjacent vertices have precisely $\lambda=\lambda(G)$ common neighbors, then $G$ is called strongly regular with parameters $(n,k,\lambda,\mu)$. A graph $G$ is called walk-regular if the number of closed walks of length $r$ from a given vertex $x$ is independent of the choice of $x$ for all $r$, that is to say, for any $x$, $A^{r}_{xx}$ is constant for all $r$, where $A$ is the adjacency matrix of $G$.

Let $X$ be a set of size $t$. The Johnson graph $J(t,d)$ $(t\geq 2d)$ is a graph with vertex set ${X\choose d}$, the set of $d$-subsets of $X$, where two $d$-subsets are adjacent whenever they have $d-1$ elements in common. $J(t,2)$ is the triangular graph $T(t)$. Recall that a clique (or a complete graph) is a graph in which every pair of vertices is adjacent. A coclique is a graph that any two distinct vertices are nonadjacent. A $t$-clique is a clique with $t$-vertices and is denoted by $K_{t}$. The line graph of $K_{t}$ is also the triangular graph $T(t)$ which is strongly regular with parameters $({t\choose 2},2t-4,t-2,4)$ and spectrum $\{(2t-4)^{1},(t-4)^{t-1},(-2)^{\frac{t^{2}-3t}{2}}\}$.

The Kronecker product $M_{1}\otimes M_{2}$ of two matrices $M_{1}$ and $M_{2}$ is obtained by replacing the $ij$-entry of $M_{1}$ by $(M_{1})_{ij}M_{2}$ for all $i$ and $j$. Note that if $\tau$ and $\eta$ are eigenvalues of $M_{1}$ and $M_{2}$ respectively, then $\tau\eta$ is an eigenvalue of $M_{1}\otimes M_{2}$.

2.2 Interlacing

Lemma 2.1** ([6], Interlacing)**

Let $N$ be a real symmetric $n\times n$ matrix with eigenvalues $\theta_{1}\geq\ldots\geq\theta_{n}$ and $R$ be a real $n\times m$ $(m<n)$ matrix with $R^{T}R=I$. Set $M=R^{T}NR$ with eigenvalues $\mu_{1}\geq\ldots\geq\mu_{m}$. Then,

1. (i)

the eigenvalues of $M$ interlace those of $N$, i.e.

[TABLE] 2. (ii)

if the interlacing is tight, that is, there exists an integer $j\in\{1,2,\ldots,m\}$ such that

$\theta_{i}=\mu_{i}$* for $1\leq i\leq j$ and $\theta_{n-m+i}=\mu_{i}$ for $j+1\leq i\leq m$, then $RM=NR.$*

In the case that $R$ is permutation-similar to \left(\begin{array}[]{cc}I&O\\ O&O\\ \end{array}\right), then $M$ is just a principal submatrix of $N$.

Let $\pi=\{V_{1},\ldots,V_{m}\}$ be the partition of the index set of the columns of $N$ and let $N$ be partitioned according to $\pi$ as

\left(\begin{array}[]{ccc}N_{1,1}&\ldots&N_{1,m}\\ \vdots&\ddots&\vdots\\ N_{m,1}&\ldots&N_{m,m}\\ \end{array}\right),

where $N_{i,j}$ denotes the block matrix of $N$ formed by rows in $V_{i}$ and columns in $V_{j}$. The characteristic matrix $P$ is the $n\times m$ matrix whose $j$th column is the characteristic vector of $V_{j}$ $(j=1,\ldots,m)$. The quotient matrix of $N$ with respect to $\pi$ is the $m\times m$ matrix $Q$ whose entries are the average row sum of the blocks $N_{ij}$ of $N$, i.e.,

[TABLE]

The partition $\pi$ is called equitable if each block $N_{i,j}$ of $N$ has constant row (and column) sum, i.e., $PQ=NP$. The following lemma can be shown by using Lemma 2.1.

Lemma 2.2

[5]* Let $N$ be a real symmetric matrix with $\pi$ as a partition of the index set of its columns. Suppose $Q$ is the quotient matrix of $N$ with respect to $\pi$, then the following hold:*

1. (i)

The eigenvalue of $Q$ interlace the eigenvalues of $N$. 2. (ii)

If the interlacing is tight (as defined in Lemma 2.1(ii)), then the partition $\pi$ is equitable.

By an equitable partition of a graph, we always mean an equitable partition of its adjacency matrix $A$.

2.3 Clique extensions of $T(n)$

In this subsection, we define $s$-clique extensions of graphs and we will give some specific results for the $s$-clique extension of triangular graphs.

Recall a $s$-clique is a clique with $s$ vertices, where $s$ is a positive integer. The $s$-clique extension of a graph $G$ with $|V(G)|$ vertices is the graph $\widetilde{G}$ obtained from $G$ by replacing each vertex $x\in V(G)$ by a clique $\widetilde{X}$ with $s$ vertices, satisfying $\widetilde{x}\sim\widetilde{y}$ in $\widetilde{G}$ if and only if $x\sim y$ in $G$, where $\widetilde{x}\in\widetilde{X},\widetilde{y}\in\widetilde{Y}$. If $\widetilde{G}$ is a $s$-clique extension of $G$, then the adjacency matrix of $\widetilde{G}$ is $(A+I_{|V(G)|})\otimes J_{s}-I_{s|V(G)|}$, where $J_{s}$ is the all-ones matrix of size $s$ and $I_{|V(G)|}$ is the identity matrix of size $|V(G)|$. In particular, if $G$ has $t+1$ distinct eigenvalues and its spectrum is

[TABLE]

then the spectrum of $\widetilde{G}$ is

[TABLE]

Note that if the adjacency matrix $A$ of a connected regular graph $G$ with $|V(G)|$ vertices and valency $k$ has four distinct eigenvalues $\{\theta_{0}=k,\theta_{1},\theta_{2},\theta_{3}\}$, then $A$ satisfies the following equation (see [7]):

[TABLE]

This implies that $G$ is walk-regular, see [10].

Now we assume $\Gamma$ is a cospectral graph with the $s$-clique extension of the triangular graph $T(n)$, where $s\geq 2$ and $n\geq 4$ are integers. Then by (2.1) and (2.2), the graph $\Gamma$ has spectrum

[TABLE]

Note that $\Gamma$ is regular with valency $k$, where $k=(s-1)+2(n-2)s=s(2n-3)-1$. Using (2.3), we obtain

[TABLE]

Therefore,

[TABLE]

The following result is known as the Hoffman bound.

Lemma 2.3

(* Cf. [2, Theorem 3.5.2]) Let $X$ be a $k$-regular graph with least eigenvalue $\tau$. Let $\alpha(X)$ be the size of maximum coclique in $X$. Then*

[TABLE]

If equality holds, then each vertex not in a coclique of size $\alpha(X)$ has exactly $-\tau$ neighbours in it.

Applying Lemma 2.3 to the complement of $\Gamma$, we obtain the following lemma.

Lemma 2.4

For any clique $C$ of $\Gamma$ with order $c$, we have

[TABLE]

If equality holds, then every vertex $x\in V(\Gamma)\setminus V(C)$ has exactly $2s$ neighbors in $C$.

3 Lines in $\Gamma$

Recall that $\Gamma$ is a graph that is cospectral with the $s$-clique extension of the triangular graph $T(n)$, where $s\geq 2$ and $n\geq 1$ are integers. This implies that $\Gamma$ is walk-regular. Now we assume that $\Gamma$ is also co-edge-regular, i.e., there exist precisely $\mu=\mu(\Gamma)$ common neighbors between any two distinct nonadjacent vertices of $\Gamma$. Note that for $\Gamma$, we have $\mu=4s$ from the spectrum of the $s$-clique extension of $T(n)$.

Fix a vertex, denoted by $\infty$ and let $\Gamma(\infty)$ be the local graph of $\Gamma$ at vertex $\infty$. Let $V(\Gamma(\infty))=\{x_{1},x_{2},\ldots,x_{k}\}$, where $k=s(2n-3)-1$. Let $x_{i}$ have valency $d_{i}$ inside $\Gamma(\infty)$ for $i=1,2,\ldots,k$. Because $\Gamma$ is walk-regular, the number of closed walks through a fixed vertex $\infty$ of length $3$ and $4$ only depend on the spectrum. This means that the number of edges in $\Gamma(\infty)$ is determined by the spectrum and as $\Gamma$ is co-edge-regular, we also see that the number of walks of length $2$ in $\Gamma(\infty)$ is determined by the spectrum of $\Gamma$. This implies these numbers are the same as in a local graph of the $s$-clique extension of $T(n)$.

Let $\Delta$ be the $s$-clique extension of $T(n)$. Fix a vertex $u$ of $\Delta$. Then there are $s-1$ vertices with valency $(s-2)+2s(n-2)$ and $2s(n-2)$ vertices with valency $s(n-2)+2(s-1)$ in the local graph of $T(n)$ with respect to a fixed vertex. Using (2.5), this implies that the sum of valencies and the sum of square of valencies of vertices in $\Gamma(\infty)$ are constant, and are given by the following equations.

[TABLE]

[TABLE]

where $\varepsilon$ is the number of edges inside $\Gamma(\infty)$. By (3.1) and (3.2), we obtain

[TABLE]

It turns out that (3.3) is of crucial importance in proving our main result. Now we show the following lemma that will be used later.

Lemma 3.1

Fix a vertex $\infty$ of $\Gamma$ and let $\Gamma(\infty)$ be the local graph of $\Gamma$ at $\infty$. Define $E:=\{y\sim\infty~{}|~{}d_{y}>\frac{3}{4}s(n-1)\}$ and let $e:=|E|$. Let $F:=\{y\sim\infty\mid d_{y}\leq\frac{3}{4}s(n-1)\}$ and $f:=|F|$. If $n\geq 55$, then the following hold.

1. (1)

$f\leq 16(s-1)$. 2. (2)

The subgraph of $\Gamma$ induced on $E$ is not complete. 3. (3)

The subgraph of $\Gamma$ induced on $E$ does not contain a coclique of order three.

Proof. Note that $f=k-e$. As $\frac{3}{4}s(n-1)+1\leq\frac{3}{4}(sn-2)$, by (3.3), we obtain

[TABLE]

So

[TABLE]

which implies $f<\frac{1}{2}(sn-2)$ if $n\geq 55$ (and $s\geq 2$). This means

[TABLE]

By Lemma 2.4, we obtain that $e$ is greater than the order of a maximum size clique and hence the subgraph induced on $E$ is not complete.

Now we show that $E$ does not contain a coclique of order three. Suppose $X\subset E$ is a coclique in $\Gamma(\infty)$ with vertices $\{x_{1},x_{2},x_{3}\}$. Define $A_{i}~{}(i=1,2,3)$ such that

[TABLE]

Since $\Gamma$ is co-edge-regular, the vertices $x_{i}$ and $x_{j}~{}(i\neq j)$ have at most $4s-1$ common neighbours. By the inclusion-exclusion principle, we have

[TABLE]

This gives $n<54$. This shows the lemma. $\blacksquare$

A maximal clique of $\Gamma$ is called a line if it contains more than $\frac{3}{4}s(n-1)$ vertices. We show the existence of lines of $\Gamma$ in the following.

Proposition 3.2

*If $n\geq 48s\geq 96$, then for every vertex $\infty$, there are exactly two lines through $\infty$, say $C_{1}$ and $C_{2}$. Denote $m:=|V(C_{1})\cap V(C_{2})\setminus\{\infty\}|$ and $\ell:=k+1-|V(C_{1})\cup V(C_{2})|$. Then $m\leq 4s-1$ and $\ell\leq 16(s-1)$. *

Proof. Fix a vertex $\infty$ of $\Gamma$, let $E=\{y\sim\infty~{}|~{}d_{y}>\frac{3}{4}s(n-1)\}$. By Lemma 3.1, a maximum coclique in $E$ has order two as $n\geq 48s\geq 55$. Let $x_{1}$, $x_{2}$ be distinct nonadjacent vertices in $E$ and let $y\in E$. Then $y$ has at least one neighbour in $\{x_{1},x_{2}\}$.

Let $A_{i}:=\{y\in E\mid y\sim x_{i},y\nsim x_{j}~{}\mbox{for}~{}~{}j=1,2,j\neq i$} for $i=1,2$. Then the subgraph induced on $A_{i}$ is complete for $i=1,2$. Let $C_{i}$ be a maximal clique containing the vertex set $\{\infty\}\cup A_{i}$ for $i=1,2$. Note that $C_{1}\neq C_{2}$ as $x_{1}\nsim x_{2}$. Let $M:=V(C_{1})\cap V(C_{2})\setminus\{\infty\}$ and $L:=V(\Gamma(\infty))\setminus(V(C_{1})\cup V(C_{2}))$. Let $m=|M|$ and $\ell=|L|$. By the co-edge-regularity of $\Gamma$, we have $m\leq 4s-1$. Let $F:=\{y\sim\infty\mid d_{y}\leq\frac{3}{4}s(n-1)\}$ and $f:=|F|$. We have, by Lemma 3.1, that $f\leq 16(s-1)$.

Suppose $x\in E\setminus(V(C_{1})\cup V(C_{2}))$. Then $x$ has at least $(\frac{3}{4}s(n-1)-(4s-2)-16(s-1))/2$ neighbours in at least one of $C_{1}$ and $C_{2}$. If $n\geq 48s\geq 96$, then this number is at least $4s$, which is a contradiction. Hence $E\subseteq V(C_{1})\cup V(C_{2})$. So, $L\subseteq F$ and hence $\ell\leq f\leq 16(s-1)$ by Lemma 3.1. This shows that $|V(C_{1})|+|V(C_{2})|\geq k-\ell\geq k-16(s-1)$. Assume $|V(C_{1})|\geq|V(C_{2})|$, then we see that

[TABLE]

as $n\geq 48s\geq 96$. This gives that there are exactly two lines through $\infty$. $\blacksquare$

Now we prove the following property for lines through a vertex.

Lemma 3.3

Fix a vertex $\infty$ of $\Gamma$ and let $C_{1}$ and $C_{2}$ be the two lines through $\infty$ with respective orders $c_{1}$ and $c_{2}$. Let $L:=V(\Gamma(\infty))\setminus(V(C_{1})\cup V(C_{2}))$ and $M:=V(C_{1})\cap V(C_{2})\setminus\{\infty\}$, and $\ell=|L|$, $m=|M|\geq 0$. If $n\geq 48s\geq 96$, then $\ell+m=s-1$ and

[TABLE]

for $i=1,2$.

Proof. Let $Q:=V(C_{1})\Delta V(C_{2})$, where $\Delta$ means “symmetric difference”. Then, by Lemma 2.4, $|Q|\leq|V(C_{1})|+|V(C_{2})|\leq 2s(n-1)$.

Note that $Q$ is the complement of $L\cup M$ inside $V(\Gamma(\infty))$.

For $y\in M$, we have

[TABLE]

by Proposition 3.2.

Now let $y\in L$. Then $y$ has at least $4s-1$ neighbors in each of $C_{1}$ and $C_{2}$. Hence, by Proposition 3.2, we obtain

[TABLE]

By (3.3), we obtain

[TABLE]

So

[TABLE]

if $n\geq 48s.$ Hence

[TABLE]

This gives for $y\in L\cup M$, using (3.5), (3.6) and $l\leq s-1$, that

[TABLE]

Note that by (3.8),

[TABLE]

for $j=1,2$.

For $y\in V(\Gamma(\infty))\setminus(L\cup M)$, we obtain

[TABLE]

Hence $|d_{y}-(sn-2)|\leq 4s.$

Now (3.3) gives us

[TABLE]

So

[TABLE]

if $n\geq 48s\geq 96.$ This implies $\ell+m=s-1$. This shows the lemma. $\blacksquare$

We obtain the following lemma immediately.

Lemma 3.4

Fix a vertex $\infty$ of $\Gamma$ and let $C_{1}$ and $C_{2}$ be the two lines through $\infty$ with respective orders $c_{1}$ and $c_{2}$. Assume $m:=|V(C_{1})\cap V(C_{2})\setminus\{\infty\}|$. If $n\geq 48s$, then $c_{1}+c_{2}=2s(n-2)+2(m+1)$.

Proof. Let $\ell=|V(\Gamma(\infty))\setminus(V(C_{1})\cup V(C_{2}))|$. Then we have

[TABLE]

If $n\geq 48s$, then we have $\ell+m=s-1$ by Lemma 3.3, hence

[TABLE]

$\blacksquare$

In the next two sections, we will follow the method as used in Hayat et al.[4].

4 The order of lines

In this section, we will show the following lemma on the order of lines.

Lemma 4.1

Let $s\geq 2$ and $n\geq 1$ be integers. Let $\Gamma$ be a co-edge-regular graph that is cospectral with the $s$-clique extension of the triangular graph $T(n)$. Let $q_{i}$ be the number of lines with order $s(n-3)+i$ for $i=1,\ldots,2s$ and $\delta=\sum_{i=1}^{2s}q_{i}$ be the number of lines in $\Gamma$. Assume $n\geq 48s$. Then

[TABLE]

holds, and the number $\delta$ satisfies

[TABLE]

If $\delta=n$, then $q_{i}=0$ for all $i<2s$, and $q_{2s}=n$.

Proof. Assume $n\geq 48s$. By Proposition 3.2, any vertex of $\Gamma$ lies on exactly two lines. Now consider the set

[TABLE]

Then, by double counting, the cardinality of the set $W$, we see (4.1). Moreover, we see that

[TABLE]

Thus, if $n>10$, we obtain

[TABLE]

On the other hand, we have

[TABLE]

This shows $\delta\geq n$, and $\delta=n$ implies that all lines have order $s(n-1)$, which means $q_{i}\neq 0$ if and only if $i=2s$. This completes the proof. $\blacksquare$

5 The neighborhood of a line

In this section we will show the following proposition.

Proposition 5.1

Let $\Gamma$ be a co-edge-regular graph that is cospectral with the $s$-clique extension of the triangular graph $T(n)$, where $s\geq 2,n\geq 1$ are integers. If $n\geq 48s$, then $\Gamma$ contains exactly $n$ lines.

Proof. In Lemma 4.1, we have seen that the number $\delta$ of lines satisfies $n\leq\delta\leq n+2$. Now we assume that $n+1\leq\delta\leq n+2$, in order to obtain a contradiction. Let $q_{i}$ be the number of lines of order $s(n-3)+i$ in $\Gamma$, where $i=1,\ldots,2s$. Let $h$ be minimal such that $q_{h}\neq 0$. Then clearly, $1\leq h\leq 2s$. Fix a line $C$ with exactly $s(n-3)+h$ vertices. Let $q^{\prime}_{i}$ be the number of lines $C^{\prime}$ with $s(n-3)+i$ vertices that intersect $C$ in at least one vertex. So $q_{i}\geq q^{\prime}_{i}$. By Lemma 3.4, we obtain

[TABLE]

By Proposition 3.2, every vertex lies on exactly two lines, and hence we obtain

[TABLE]

Now multiply (5.2) by 2 and subtract (4.1) from obtained equation, we find

[TABLE]

as $\delta=\sum\limits_{i=1}^{2s}q_{i}$. This gives

[TABLE]

As $n+1\leq\delta\leq n+2$, we see

[TABLE]

Since $n\geq 48s$, (5.4) implies that $h\geq 3s$. This contradicts to $h\leq 2s$. This completes the proof. $\blacksquare$

6 Proof of the main result

In this section we show our main result, Theorem 1.1.

Proof of Theorem 1.1. Assume $n\geq 48s$. By Propositions 3.2 and 5.1 and Lemma 4.1, we find that there are exactly $n$ lines, each of order $s(n-1)$, and every vertex $x$ in $\Gamma$ lies on exactly two lines. Moreover, by Lemma 3.4, the two lines through any vertex $x$ have exactly $s$ vertices in common, and every neighbor of $x$ lies in one of the two lines through $x$. Now consider the following equivalence relation $\mathcal{R}$ on the vertex set $V(\Gamma)$: $x\mathcal{R}x^{\prime}$ if and only if $\{x\}\cup N_{\Gamma}(x)=\{x^{\prime}\}\cup N_{\Gamma}(x^{\prime})$, where $x,x^{\prime}\in V(\Gamma)$.

Every equivalence class under $\mathcal{R}$ contains $s$ vertices and it is the intersection of two lines. Let us define the graph $\hat{\Gamma}$ whose vertices are the equivalent classes and two classes, say $S_{1}$ and $S_{2}$, are adjacent in $\hat{\Gamma}$ if and only if any vertex in $S_{1}$ is adjacent to any vertex in $S_{2}$. Then $\hat{\Gamma}$ is a regular graph with valency $2n-4$, and $\Gamma$ is the $s$-clique extension of $\hat{\Gamma}$. Note that the spectrum of $\hat{\Gamma}$ is equal to

[TABLE]

by the relation of the spectra of $\Gamma$ and $\hat{\Gamma}$, see (2.1) and (2.2). Since $\hat{\Gamma}$ is a connected regular graph with valency $2n-4$, and it has exactly three distinct eigenvalues, it follows that $\hat{\Gamma}$ is a strongly regular graph with parameters $(\binom{n}{2},2n-4,n-2,4)$.

As proved in [3], the triangular graphs are determined by the spectrum except when $n=8$, but we suppose $n$ is large enough then the assertion follows immediately. This completes the proof. $\blacksquare$

7 Acknowledgements

Jack Koolen is partially supported by the National Natural Science Foundation of China (No. 11471009 and No. 11671376) and Anhui Initiative in Quantum Information Technologies (No. AHY150000). Ying-Ying Tan is supported by the National Natural Science Foundation of China (No. 11801007) and Natural Science Foundation of Anhui Province (No. 1808085MA17) and Doctoral Start-up foundation of Anhui Jianzhu University (No. 2018QD22). Zheng-jiang Xia is supported by the University Natural Science Research Project of Anhui Province (No. KJ2018A0438).