Distance-regular graphs without 4-claws
Sejeong Bang, Alexander Gavrilyuk, Jack Koolen

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 and but without induced -subgraphs.
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Coding theory and cryptography
Distance-regular graphs without -claws
**Sejeong Bang
**Department of Mathematics
Yeungnam University, Gyeongsan-si, Gyeongbuk 38541, Republic of Korea
e-mail: [email protected]
**Alexander Gavrilyuk
**School of Mathematical Sciences
University of Science and Technology of China, Hefei, 230026, Anhui, PR China;
N.N. Krasovskii Institute of Mathematics and Mechanics,
Ural Branch of Russian Academy of Sciences, Yekaterinburg, Russia
e-mail: [email protected]
**Jack Koolen
**School of Mathematical Sciences
University of Science and Technology of China, Hefei, 230026, Anhui, PR China
e-mail: [email protected]
Abstract
We determine the distance-regular graphs with diameter at least and but without induced -subgraphs.
1 Introduction
A -claw is a complete bipartite graph with parts of size and . We say that a graph is -claw-free (or without -claws), if it does not contain a -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 -claw-free graphs as a generalization of line graphs (it is easy to see that line graphs are -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 -claw-free graph can be obtained from one of the so-called basic -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 . In fact, any connected graph has smallest eigenvalue at most 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 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 is regular with valency , then its line graph is regular with valency , and it contains a set of maximal cliques of the same order equal to such that every edge of belongs to a unique member of .
For distance-regular graphs, this observation can be formulated in a more general way. Let be a distance-regular graph with valency , diameter and smallest eigenvalue . Then any clique of 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 in is called a Delsarte clique, if contains exactly vertices. Note that, if contains a Delsarte clique, then must be integral as is integral.
Godsil [22] introduced the following notion of a geometric distance-regular graph: a non-complete distance-regular graph is called geometric, if there exists a set of Delsarte cliques such that every edge of lies in a unique member of . In this case we say that is geometric with respect to . 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 , there are only finitely many coconnected non-geometric distance-regular graphs with smallest eigenvalue at least and diameter . Koolen and Bang [27] generalised this as follows: there are only finitely many coconnected non-geometric distance-regular graphs with smallest eigenvalue at least and with given diameter or having the intersection number (in fact, the Bannai-Ito conjecture recently proved by Bang et al. [5] shows that the latter condition on 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 , any geometric distance-regular graph with smallest eigenvalue , diameter and 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 .
Note that a geometric distance-regular graph with smallest eigenvalue is -claw-free, since every edge is contained in exactly one Delsarte clique, and hence each vertex is contained in exactly Delsarte cliques.
On the other hand, Godsil [22] showed that is geometric, if is -claw-free for any and the intersection numbers and of satisfy the following inequality:
[TABLE]
Koolen and Bang [27] showed that is geometric, if holds. Moreover, they [19, Proposition 9.7] showed that a distance-regular graph of diameter having valency for some integer , is geometric with smallest eigenvalue if and only if is -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 -claws. Hiraki, Nomura and Suzuki [24] and Yamazaki [38] considered distance-regular graphs that are locally a disjoint union of three cliques of size (i.e., 4-claw-free), and these graphs for turn out to be geometric with smallest eigenvalue (see Remark 4.6). The problem of classification of such graphs seems to be very hard. Hence we may assume . Bang [4] showed that if , then a distance-regular graph of valency is -claw-free if and only if is geometric with smallest eigenvalue . In [4], Bang started the classification of geometric distance-regular graphs with smallest eigenvalue and , and the results of Bang and Koolen [7] and Gavrilyuk and Makhnev [21] completed the classification.
Our main result is the following classification of -claw-free distance-regular graphs.
Theorem 1.1
Let be a distance-regular graph with diameter , valency but without -claws. If , then is one of the following graphs:
- (1)
The Taylor graph with with ;
- (2)
The Klein graph with ;
- (3)
The Gosset graph with ;
- (4)
The halved -cube with with ;
- (5)
The Hamming graph with ;
- (6)
The Johnson graph with ;
or is a putative distance-regular graph with one of the following intersection arrays:
- (7)
;
- (8)
;
- (9)
.
The proof of Theorem 1.1 is organised according to the following three cases with respect to the valency and the intersection number of a distance-regular graph satisfying the condition of the theorem:
- (T)
,
- (N)
,
- (G)
.
If satisfies case (T), then it is either the Johnson graph or the halved -cube or a Taylor graph (see Proposition 2.1). In Section 3, we classify the Taylor graphs without -claws.
Now we consider the distance-regular graphs that satisfy cases (G) or (N). Here we show that the diameter of is at most 4 (see Lemma 2.10), and the smallest eigenvalue of is bounded below in terms of the ratio (see Lemma 2.7).
If satisfies case (G), then it was essentially shown by Bang (see Proposition 2.11) that is -claw-free if and only if it is geometric with smallest eigenvalue . If is geometric with case (N), then we observe that there are only finitely many possibilities for , and this enables us to classify all geometric distance-regular graphs with but without -claws in Section 4.
In Section 5, we consider non-geometric distance-regular graphs with case (N). Since the smallest eigenvalue of 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 is bounded above as , where is the multiplicity of second largest eigenvalue. However, their bound for is too large, which makes the search for feasible intersection arrays for almost intractable. In Subsection 5.1, we obtain a better valency bound of order by using that any clique size gives a lower bound for (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 -claws. Moreover, in Proposition 5.20, we show that there does not exist a distance-regular graph with intersection array .
Some of our results can be possibly generalised for distance-regular graphs without -claws for any , 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 -claws.
Note that the complement of a -claw-free strongly regular graph does not contain any -clique. In general, it appears to be hard to show the non-existence of -cliques in a strongly regular graph by using only its parameters (for example, the Shrikhande graph and the -grid both have the same parameters, but the latter one contains a -clique, while the former one does not; see further [12] for the case ). This suggests that the assumption 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 , the distance between any two vertices is the length of a shortest path between and in , and the diameter is the maximum distance ranging over all pairs of vertices of .
For any vertex of , let be the set of vertices in at distance precisely from , where . For a non-empty subset , denotes the induced subgraph on . The local graph of a vertex is . For a class of graphs, a graph is called locally , if, for any vertex of , its local graph is isomorphic to a graph from . If all graphs from are isomorphic to a graph , we say that is locally . A graph is regular with valency , if contains precisely vertices for all .
For a set of vertices of , let denote . Recall that a clique (coclique, resp.) is a set of pairwise adjacent (non-adjacent, resp.) vertices. A connected non-complete graph is called Terwilliger, if for any two vertices at distance , is a clique, whose order does not depend on the choice of and .
A connected graph with diameter is called distance-regular, if there exist integers , ) such that, for any two vertices with , there are precisely neighbors of in and neighbors of in . Set . The numbers and are called the intersection numbers of . We observe and . The array
[TABLE]
is called the intersection array of . In particular, is a regular graph with valency . We further note that, for , the number does not depend on the choice of a vertex of . Define for any vertex and . Then we have and thus
[TABLE]
so that .
A distance-regular graph with diameter 2 is called a strongly regular graph. We say that a strongly regular graph has parameters , where , and .
Let be a distance-regular graph of diameter . The adjacency matrix is the -matrix with rows and columns indexed by , where the -entry of is , if , and [math] otherwise. The eigenvalues of are the eigenvalues of . It is well-known that a distance-regular graph with diameter has exactly distinct eigenvalues , which are the eigenvalues of the following tridiagonal matrix (cf. [16, p.128]):
[TABLE]
The standard sequence corresponding to an eigenvalue is a sequence satisfying the following recurrence relation:
[TABLE]
where and . Then the multiplicity of eigenvalue is given by
[TABLE]
which is known as Biggs’ formula (cf. [9, Theorem 21.4], [16, Theorem 4.1.4]). Let denote the multiplicity of eigenvalue . Then
[TABLE]
where is the trace of matrix (cf. [9, Lemma 2.5]).
Let and 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 be a distance-regular graph of diameter , and with intersection array given by Eq. (2), valency , and distinct eigenvalues with multiplicities , , , , respectively.
Proposition 2.1
([28, Theorem 16]) If and , then 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 -gon of order for some ;
- (4)
A Taylor graph;
- (5)
The Johnson graph ;
- (6)
The halved -cube.
Lemma 2.2
If and is not a Terwilliger graph, then holds (for all ) and
[TABLE]
*Proof: *The inequality on follows from [16, Theorem 5.2.1]. Suppose that is an induced quadrangle where . Put for . Then . As , there exists a vertex and we find
[TABLE]
Hence
[TABLE]
which shows the result.
The following results relate eigenvalues and intersection numbers of .
Theorem 2.3
([30, Theorem 3.6]) The following inequality holds:
[TABLE]
with equality if and only if .
Lemma 2.4
([30, Proposition 3.2]) If , then the following holds.
- (1)
.
- (2)
.
- (3)
* lies between and .*
Lemma 2.5
*If , then , and implies . *
*Proof: *(1) Suppose . For any two vertices with , the subgraph is the disjoint union of two connected regular graphs with valency . By the interlacing result (see [16, Theorem 3.3.1]) we have and , a contradiction.
(2) By (1) and Lemma 2.4 (2), we find that and . Hence the result follows by .
The following result gives some bounds on the multiplicity of the second largest eigenvalue of .
Lemma 2.6
(cf. [34, Theorem 3], [16, Theorem 4.4.4]) Suppose that holds.
- (1)
If has a clique of size , then
[TABLE]
- (2)
If is irrational with multiplicity , then
[TABLE]
*Proof: *(1) It is straightforward by [34, Theorem 3].
(2) By [16, Theorem 4.4.4], and are conjugate and thus . Moreover, each local graph of has eigenvalues and with multiplicities at least and , respectively. This shows that , and hence we find .
2.3 Some structure theory for distance-regular graphs without -claws
In this subsection, we give some results on distance-regular graphs without -claws ().
Lemma 2.7
Suppose that contains a -claw but no -claws for some integer . With , the following holds.
- (1)
* contains a clique with at least vertices. In particular,*
[TABLE]
- (2)
.
*Proof: *By assumption, contains a -claw, say , where .
(1) For each , we define a set
[TABLE]
We first show the following claim.
Claim 2.8
There is an integer such that .
*Proof of Claim 2.8: * Let be the number of edges between and . As each vertex in has exactly neighbors in , we have . Since each vertex in has a unique neighbor in and each vertex in has at least two neighbors in , we find . Hence it follows by and that
[TABLE]
and thus . Now the claim follows as .
Without loss of generality, let satisfy by Claim 2.8. If there exist two non-adjacent vertices , then is a -claw, which is impossible. So induces a clique and thus
[TABLE]
follows by Eq. (1). Now we have
[TABLE]
and this shows (1).
(2) Since there are no -claws in , we have . 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 . Define by
Corollary 2.9
Suppose that contains a -claw but no -claws. Then there exists a clique of size at least .
*Proof: *Let be a set, which induces a -claw, where . Put . As there are no -claws, holds. Thus, follows by
[TABLE]
Since there are no -claws in , we see that induces a clique, and hence induces a clique of size , which proves the claim.
Lemma 2.10
Suppose that contains a -claw but no -claws. If and , then the following holds.
- (1)
* contains an induced quadrangle (i.e., ). *
- (2)
.
- (3)
.
- (4)
If , then .
*Proof: *(1) As contains a -claw and , it follows from Lemma 2.7 (2) that . Suppose that has no induced quadrangles (i.e., is a Terwilliger graph). Then, by [28, Proposition 6 (3)], is the icosahedron, the Conway-Smith graph, or the Doro graph. Both the Conway-Smith graph and the Doro graph have -claws, and the icosahedron does not have a -claw, a contradiction.
(2) By (1) there is a quadrangle, say , where . As and is -claw-free, must be a clique of size . This shows that and . Since is a clique of size , the result follows by Eq. (1).
(3) Suppose . By (1), there is a quadrangle, say with . As , there is a vertex satisfying and . As is a coclique in and is -claw-free, the set induces a clique of size . It follows by (2), Lemma 2.2, and that and thus
[TABLE]
Let and take . As , there is a vertex , and take a vertex . Similarly, there is a vertex as . Take a vertex . As the set induces a clique, for any , two vertices and has common neighbors in and thus is a coclique. On the other hand, there exists a vertex by , and the set induces a coclique as () and . Hence is a -claw in , which is impossible. This shows .
(4) As holds by (2), the result (4) follows from Lemma 2.4 (3).
Proposition 2.11
(cf. [4, Theorem 3.2]) If , then the following statements are equivalent.
- (1)
* is -claw-free.*
- (2)
* is a geometric distance-regular graph with smallest eigenvalue .*
*Proof: *If , then and . Hence is the cube (see [10]), which satisfies both and . If , then the result follows from [4, Theorem 3.2].
3 Taylor graphs without -claws
In this section we classify the Taylor graphs without -claws (Theorem 3.1).
Let be a Taylor graph (i.e., a distance-regular graph with diameter three and intersection array ). The eigenvalues of are
[TABLE]
whose multiplicities are given by
[TABLE]
Theorem 3.1
Let be a Taylor graph without -claws. Then is one of the following graphs:
- (1)
The hexagon;
- (2)
The cube with ;
- (3)
The icosahedron with ;
- (4)
The Johnson graph with ;
- (5)
The Taylor graph with ;
- (6)
The halved -cube with ;
- (7)
The Taylor graph with ;
- (8)
The Gosset graph with .
Moreover, the hexagon, the cube and the Johnson graph are the only geometric Taylor graphs without -claws.
*Proof: *If is -claw-free, then is either the hexagon or the icosahedron by [11, Theorem 1.2]. We assume that is a Taylor graph with -claws but without -claws. If , then is the cube (see [16, Theorem 7.5.1]). From now on, we assume . If , then it follows by Eq. (5) and Eq. (7) that and , which implies that is bipartite. This is impossible as and (i.e., contains -claws). Hence . As , holds. For a vertex , let be the local graph of . By [16, Theorem 1.5.3], is a strongly regular graph with parameters and eigenvalues , where and
[TABLE]
By Eqs. (1), (6) and Corollary 2.9,
[TABLE]
If , then and follow by Eqs. (6), (7) and in Eq. (5). As holds by Eq. (5), we find (thus is odd) and hence holds as
[TABLE]
by Eq. (9). On the other hand, is even and is a sum of two squares by and [16, Proposition 1.10.5 (i)]. These restrictions imply . If , then is the icosahedron, which is -claw-free. If , then is the Johnson graph which contains -claws but no -claws. If , then is the Payley graphs and , respectively. As the independence numbers of these Payley graphs are all three, contains -claws but does not contain -claws. If , then each local graph of is a strongly regular graph with parameters . Since all such stongly regular graphs contain cocliques (see [33]), contains -claws and thus . Now the results (4), (5) and (7) follow.
Now we assume that . Then all the eigenvalues are integral. As and by [16, Theorem 4.4.4], holds for with , and
[TABLE]
is the smallest eigenvalue of , i.e., . It follows by Eq. (8) with that
[TABLE]
By Eqs. (9), (10) and (11), we find that , which shows . If , then and thus with and by Eq. (6). As holds by Eq. (7), we find as . By [16, Corollary 1.15.3], . If , then is the halved -cube whose local graph is triangular graph and independence number of is three. If , then is the Gosset graph whose local graph is the Schläfli graph and its independence number is also three. If , then it follows by Eq. (11) that holds and thus , and by Eqs. (6) and (7). As , , and is even, . Note that (the complement of local graph ) is also a strongly regular graph with parameters . It follows by [12, Theorem 3.3] and [13] that for each , contains cliques of size . This implies contains cocliques of size and thus does contain -claws, a contradiction. If , then is the McLaughlin graph which has a coclique of size , this is also impossible. This completes the proof.
4 Geometric distance-regular graphs without -claws
In this section we classify geometric distance-regular graphs with diameter but without -claws (see Theorem 4.5).
Let be a geometric distance-regular graph with respect to a set of Delsarte cliques and diameter . For a vertex and a non-empty subset of , we denote . For a Delsarte clique and for each , let . For a vertex , define
[TABLE]
For with , define
[TABLE]
Since parameters and depend only on the distances and respectively (see [6, Lemma 4.1] and [23, Section 11.7]), we denote and . We observe that for any vertex and for any Delsarte clique ,
[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 be a geometric distance-regular graph with diameter . Then the following holds.
- (1)
.
- (2)
.
- (3)
* if and only if .*
- (4)
If , then .
Lemma 4.2
([19, Proposition 9.9]) Let be an integer, and let be a distance-regular graph with diameter and valency . If , then is geometric with .
Lemma 4.3
Let be a geometric distance-regular graph with -claws but without -claws. Then either with or with .
*Proof: *As is -claw-free, . It follows by Lemma 2.10 (1) that either with or with . Let satisfy and . Then it is enough to show . We first show the following claim.
Claim 4.4
For any vertices and with , there is a quadrangle containing and .
Proof of Claim 4.4: Let . As is geometric, there is a Delsarte clique satisfying and . As , we have by Lemma 4.1 (3) and thus there exists a vertex . By , and , there exists a vertex such that and thus . Hence the induced subgraph on is a quadrangle.
If , then there exist vertices satisfying and . By Claim 4.4, there are vertices and such that and are quadrangles, respectively. This is impossible as the set induces a -claw. This shows .
Now we prove the main result of this section.
Theorem 4.5
Let be a geometric distance-regular graph with diameter , valency but without -claws. Then is one of the following graphs:
- (1)
One of the two generalized hexagons of order with ;
- (2)
The generalized hexagon of order with ;
- (3)
The halved Foster graph with ;
- (4)
A generalized octagon of order with ;
- (5)
The Hamming graph with ;
- (6)
The Johnson graph with ;
- (7)
The line graph of a Moore graph;
- (8)
The flag graph of a regular generalized -gon of order for some ;
- (9)
A distance-regular graph with satisfying , and .
*Proof: *If is -claw-free, then it follows by [11, Theorem 1.2] that is (7) or (8). (Note that these graphs are all geometric as they satisfy and , and the icosahedron is non-geometric as it has irrational smallest eigenvalue.) Now we suppose that contains -claws. If , then it follows by Proposition 2.1 and Theorem 3.1 that is the Johnson graph with . If , then it follows by Proposition 2.11, [4, Theorem 4.3] and [7, Theorem 1.4] that is one of the following:
(a) One of the two generalized hexagons of order with ;
(b) The generalized hexagon of order with ;
(c) The halved Foster graph with ;
(d) A generalized octagon of order with ;
(e) The Hamming graph with ;
(f) The Johnson graph with ;
(g) A distance-regular graph with satisfying , and .From now on, we assume that . Note that and follow by Lemma 4.3. Since there is a quadrangle by Lemma 2.10 (1), there are two vertices and at distance two such that vertices induce a quadrangle for some . As , there exists . This implies that the set induces a -claw and thus by Eq. (12) and Lemma 2.7 (1). If then , which is impossible as and . Thus . If , then is the Johnson graph with by [4, Theorem 4.3]. By Eq. (12) with , there are at least four Delsarte cliques containing a fixed vertex . If , then there exist vertices and such that the set induces a -claw, which is impossible and hence
[TABLE]
It follows by with Eq. (13) that if , then
[TABLE]
This is impossible as . Hence and thus and follow by Lemma 4.1 (4) and Eq. (13). Using Eq. (13), and (see [6, Lemma 5.2 (ii)]) we find By the feasible conditions , Eq. (5), Lemma 4.1 and (see [6, Theorem 5.5]), we find that the feasible intersection arrays for are , and . 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 satisfying and (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 satisfying , and ) is one of the following:
(1) A distance-regular graph satisfying and*
[TABLE]
(2) A distance-regular graph satisfying and
[TABLE]
(3) A halved graph of a distance-biregular graph with vertices of valency and
[TABLE]
It is unknown whether there exists a constant such that any distance-regular graph with and has diameter .
Question 4.7
Is it true that any geometric distance-regular graph with smallest eigenvalue has a -claw?
5 Non-geometric distance-regular graphs without -claws
In this section, we classify non-geometric distance-regular graphs with diameter and valency but without -claws (see Theorem 5.1).
Theorem 5.1
Let be a non-geometric distance-regular graph with diameter , valency but without -claws. Then is one of the following graphs:
- (1)
the icosahedron with ;
- (2)
the Klein graph with ;
- (3)
the halved -cube with with ;
- (4)
the Taylor graph with with ;
- (5)
the Gosset graph with ;
or has one of the following intersection arrays:
- (6)
;
- (7)
;
- (8)
.
To prove Theorem 5.1, we first show in Subsection 5.1 that the valency of 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 and -claws but without -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 -claws. The proof of Theorem 5.1 is given in Subsection 5.3.
We first need the following lemma.
Lemma 5.2
If is a non-geometric distance-regular graph with diameter , valency , and -claws but without -claws, then and hold. Moreover, if , then is one of the following graphs:
- (1)
the halved -cube with with ;
- (2)
the Taylor graph with with ;
- (3)
the Gosset graph with .
*Proof: *If , then and is geometric by [4, Theorem 3.2]. As is non-geometric and , it follows from Proposition 2.11 that holds. Suppose that holds. By Proposition 2.1, the graph is the halved -cube or a Taylor graph. In the latter case, the result follows from Theorem 3.1.
For the rest of this section, let be a non-geometric distance-regular graph with diameter , valency , and -claws but without -claws. By Lemma 5.2, it follows that , and, if , then . Thus, in what follows, we assume that the valency satisfies
[TABLE]
Further, let be the distinct eigenvalues of with multiplicities , , , , respectively. Lemma 2.10 (3) shows that . By Lemma 2.7 (1) (with and ), holds.
5.1 Valency bounds
It is known that for given integer , there are only finitely many non-geometric distance-regular graphs with both valency and diameter at least and with smallest eigenvalue at least , 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 of a non-geometric distance-regular graph with diameter but without -claws.
To obtain a valency bound, we consider the following two cases: and . In the latter case, we also consider several subcases depending on ( or ) and ( or ).
The main result of this subsection is the following proposition.
Proposition 5.3
*Let be a non-geometric distance-regular graph with diameter , valency , and -claws but without -claws. Then one of the following holds. *
- (1)
* holds, , and:*
- (i)
if , then and , 2. (ii)
if , then and . 2. (2)
* holds, , and:*
- (i)
if , then , 2. (ii)
if and , then , 3. (iii)
if , and , then , 4. (iv)
if , and , then .
To prove Proposition 5.3, we need several lemmas.
Lemma 5.4
If and , then .
*Proof: *If and is integral, then it follows by [16, Theorem 4.4.4] that divides , i.e., holds for some integer .
By Eq. (14), . As , we find and thus . As is non-geometric and it satisfies , with , and (see Lemma 5.2), it follows by [16, Theorem 4.4.11] that there are no such graphs. Thus , and the lemma is proved.
Lemma 5.5
If and both hold for some constant , then
[TABLE]
*Proof: *Suppose that and hold for some constant . It follows by Eq. (4), and that
[TABLE]
and hence
[TABLE]
We also have that:
- •
by Eq. (14),
- •
by Lemma 2.10 (2),
- •
holds by Lemma 4.2 as is not geometric,
- •
by Eq. (14) and Lemmas 2.2 and 2.10 (2),
- •
if , then by Eq. (14) and Lemmas 2.2 and 2.10 (2).
Suppose so that . Then we find by Eq. (3) that
[TABLE]
and hence
[TABLE]
Note that the function is strictly increasing on , therefore
[TABLE]
Suppose . Then and thus . It follows by Eq. (15) that and thus
[TABLE]
which completes the proof.
The following technical lemma will be used later (see Lemma 5.7).
Lemma 5.6
If and for some constant , then and
[TABLE]
*Proof: *Suppose that and hold for some constant . By Lemma 2.5, holds. Using Eq. (5) we have
[TABLE]
Note that by [16, Theorem 4.4.4] and thus .
By Lemma 2.10 (4), holds and hence . It follows by Eq. (16) that
[TABLE]
Now the result follows as holds by Theorem 2.3.
Lemma 5.7
If and both hold, then .
*Proof: *By Eq. (14) and Lemmas 2.5, 2.10 (2) and 4.2, we have that
[TABLE]
Claim 5.8
The valency satisfies
[TABLE]
*Proof of Claim 5.8: * If , then . Now we assume . We first show . By Eq. (17), we find
[TABLE]
and thus .
As holds by Lemma 2.7, we find by Theorem 2.3. It follows by Lemma 5.6 that
[TABLE]
and thus . As , we find and hence . Now the claim follows.
Evaluating the valency bound from Claim 5.8 for shows that holds, and this completes the proof.
Remark 5.9
Lemma 2.6 together with Lemma 2.7 show that the multiplicity can be bounded below by with an appropriate constant . Therefore the valency 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 for the case when and .
Lemma 5.10
Suppose that , , and . Then .
*Proof: *It follows from Lemma 2.5 and [16, Theorem 4.4.4] that and , and thus, by Lemma 2.10 (4), is integral satisfying .
We first show that if , then . If , then holds by [8, Theorem 4.2]. Let . It follows by Eq. (14) and [8, Theorem 4.2 (iv)] that
[TABLE]
where . It follows from Eq. (18) that , and thus holds, if .
Now we assume that . We will show that . If follows from Eq. (5), and that
[TABLE]
and hence and . Thus, we have that and .
If , then it follows by , and (see Lemma 2.6 (2)) that
[TABLE]
Hence . This shows that if . This completes the proof.
Lemma 5.11
If and , then and hold for some . Moreover,
[TABLE]
*Proof: *Suppose that is integral with multiplicity . By Lemma 5.4, we have , and, by Lemma 2.5, we have .
By [16, Theorem 4.4.4], holds for some integer , and is an eigenvalue of the local graph for a vertex . As holds by Eq. (14), it follows by Lemma 2.7 that and thus as follows by the interlacing result (see [16, Corollary 3.3.2]). Note that if , then by [16, Theorem 4.4.11] is the halved -cube with , or the Gosset graph, i.e., satisfies , a contradiction.
Suppose . By Eq. (14), holds. By Lemma 2.4 (1), we have and thus
[TABLE]
(Note here that holds from .)
By Lemma 2.4 (2) with , we have , and thus . By Lemma 2.10, we have
[TABLE]
Hence it follows by and Eq. (19) that
[TABLE]
Now the result follows as . In particular, using Eq. (19) and ,
[TABLE]
This shows .
Lemma 5.12
Suppose that , , and . If the smallest eigenvalue of is not integral, then .
*Proof: *By Lemma 2.5, we have . Suppose that . Then the eigenvalues and are conjugate algebraic integers with as . By Eq. (5), we have
[TABLE]
This shows
[TABLE]
By Lemma 5.11, holds for some . In particular,
[TABLE]
holds for some . If , then as .
In the rest of the proof, we assume and . It follows by Lemma 2.10 (4) that . We find by Eqs. (20) and (21) that
[TABLE]
[TABLE]
follows and thus
[TABLE]
We now put , where
[TABLE]
Then
[TABLE]
To determine an upper bound in Eq. (24), we first show the following claim.
Claim 5.13
*Let and be some real numbers.
(1) If holds, then .
(2) If holds, then .*
*Proof of Claim 5.13: * (1) For given and , function of is decreasing on as
[TABLE]
Thus is also decreasing on , and this shows part (1) as .
(2) As , it is enough to show that holds for all . Note that . By Eq. (23), we have
[TABLE]
If then holds by Eq. (5.1). Now we suppose that holds. Then for each , 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 , then it follows by Lemma 2.7 (1) that there exists a clique satisfying and thus . By Eq. (24) and Claim 5.13, we find as
[TABLE]
It follows by Lemma 2.6 (1) that and thus .
Now we suppose that . By Lemma 2.7 (1), there exists a clique satisfying and thus . Now we find as
[TABLE]
holds by Eq. (24) and Claim 5.13. Hence it follows by Lemma 2.6 (1) that and thus .
This shows that if , then . Recall that , if . This completes the proof.
Lemma 5.14
Suppose that , and . If the smallest eigenvalue of is integral, then .
*Proof: *By Eq. (14), Lemmas 5.2 and 5.11, we have that for some number , for some , and .
Using Eq. (5), we have Note that by [16, Theorem 4.4.4] and thus . As holds by Lemma 2.10 (4), we obtain that
[TABLE]
By Lemma 5.11, and , it follows that
[TABLE]
To complete the proof of the lemma, we consider the following three cases.
If , then it follows by Lemma 2.7 (1) that there exists a clique satisfying and thus . Hence it follows by Lemma 2.6 (1) and Eq. (26) with that . This shows .
If , then it follows by Lemma 2.7 (1) that there exists a clique satisfying and thus . Hence it follows by Lemma 2.6 (1) and Eq. (26) with that . This shows .
If , then it follows by Lemma 2.7 (1) that there exists a clique satisfying and thus . Hence it follows by Lemma 2.6 (1) and Eq. (26) with that . This shows .
This completes the proof.
*Proof of Proposition 5.3: * By Lemma 2.10 (3), holds.
(1) Suppose that holds. If , then holds by Lemma 5.4, and thus by Lemma 5.5. If , then holds by Lemma 2.6 (2), and thus by Lemma 5.5. This shows (1).
(2) Suppose that . Then 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 and -claws but without -claws whose valency satisfies the bounds in Proposition 5.3.
We call an intersection array feasible, if it satisfies the following conditions:
;
(the handshake lemma);
(see [16, Lemma 4.3.1]), where ;
all multiplicities calculated by Eq. (4) are positive integers;
all intersection numbers are non-negative integers.
Suppose that is a non-geometric distance-regular graph with diameter , valency , and -claws but without -claws.
For each of the following cases given by Proposition 5.3:
, , , , and ; 2.
, , , and ; 3.
, , , and ; 4.
, , , , and ; 5.
, , , , , and ; 6.
, , , , , and ,
we check by computer all feasible intersection arrays additionally satisfying:
- •
(see Lemma 5.2),
- •
(by Lemma 2.7 with and ),
- •
(see Lemma 2.10 (1) and Lemma 4.2), and
- •
(by Lemma 2.10 (2)).
For a given intersection array, we successively check the properties , . If it passes –, we then check by calculating the eigenvalues and their multiplicities. For both cases, and , we apply a root-finding algorithm known as Brent’s method [15] for the characteristic polynomial of the matrix . For , if we find one eigenvalue then we apply the observation in [16, p. 130] in order to calculate two remaining non-trivial eigenvalues. For , we look for two roots in the intervals and , respectively (as by [16, Proposition 4.4.9(i)] and by the assumption). If two such roots are found, then we proceed as in the case . Finally, if all the multiplicities , are integral and positive, then we check whether holds.
We can now summarize the computational results.
Proposition 5.15
The following holds.
- (1)
The only feasible intersection arrays satisfying are:
[TABLE] 2. (2)
The only feasible intersection array satisfying is:
[TABLE] 3. (3)
The only feasible intersection arrays satisfying are:
[TABLE] 4. (4)
There are no feasible intersection arrays satisfying . 5. (5)
There are no feasible intersection arrays satisfying . 6. (6)
The only feasible intersection arrays satisfying are:
[TABLE]
[TABLE]
[TABLE]
5.3 Non-existence of some distance-regular graphs without -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 -claws.
Lemma 5.16
There are no -claw-free distance-regular graphs with intersection arrays satisfying :
[TABLE]
*Proof: *The intersection array 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 and , 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 -claw-free distance-regular graphs with the following intersection arrays satisfying :
[TABLE]
[TABLE]
*Proof: *The intersection arrays , , , and satisfy and . This is impossible as follows from Lemma 2.7 (1) (see the proof of Lemma 5.14).
A distance-regular graph with intersection array has eigenvalues and with multiplicities and by Eq. (4). It follows by [16, Theorem 4.4.4] that each local graph of has eigenvalues with multiplicity and with multiplicity . Hence each local graph is a strongly regular graph with parameters , i.e. the unique generalized quadrangle . Since is the complement of the triangular graph , which has a 4-clique, contains a 4-coclique and thus has a 4-claw. This is impossible.
Remark 5.18
We note that there exists a distance-regular graph with intersection array . It was constructed in [26]. One can check that this graph contains -claws, however, it is an open question whether a distance-regular graph with this intersection array must have a -claw or not.
In the rest of this subsection, we study this problem for the intersection arrays satisfying . We note that there exists a distance-regular graph with intersection array , given by the Mathon construction, see [16, Proposition 12.5.3]. There also exists a distance-regular graph with intersection array , which is locally folded 5-cube, see [16, p. 386, Remark (iii)].
Lemma 5.19
Any distance-regular graph with intersection array or contains a -claw.
*Proof: *Suppose that is a -claw-free distance-regular graph with intersection array or . Note that for both possibilities.
The Delsarte bound (see Eq. (1)) for shows that a clique in contains at most vertices.
Pick a vertex , and let be two non-adjacent vertices of . As is -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 , where . For , we have . Thus .
Now pick a vertex from Y:=\Gamma_{1}(x)\setminus\big{(}\{y_{1},y_{2}\}\cup\Gamma_{1}(y_{1})\cup\Gamma_{1}(y_{2})\big{)}. As holds for any , we see that . This is impossible, as , and the lemma follows.
Proposition 5.20
There are no distance-regular graphs with intersection array .
*Proof: *Let be a distance-regular graph with intersection array . Then , , , , and , , , .
For a vertex , if the local graph contains an -coclique then
[TABLE]
holds by [29, Lemma 2]. This yields that is -claw-free, as .
Let induce a -coclique in . As is -claw-free, we see that
[TABLE]
so that . This implies that lies in a maximal clique of , say , with vertices, . The Delsarte bound (see Eq. (1)) for shows that .
The number of vertices of is equal to
[TABLE]
so that .
Let be the set of maximal cliques in with at least vertices. Then , and for every vertex , there exists a clique , . As , we have for all distinct . Therefore satisfies one of the following:
- (a)
, where every is a Delsarte clique in ;
- (b)
, where , , , and and intersect in exactly three vertices;
- (c)
, where , , , and and intersect in exactly three vertices.
Claim 5.21
Case (b) is impossible.
Proof of Claim 5.21: In this case there exists a vertex with 6 neighbours in ( has valency 14, and has no neighbours in ), but there are at most three of its neighbours in and at most three of its neighbours in . This implies that has at least four neighbours in , which is a Delsarte clique in , a contradiction. This shows the claim.
Thus, by Claim 5.21 and the connectivity of , for all vertices in , the sets of maximal cliques in with at least vertices satisfy either Case or Case from the above.
In Case (a), is geometric, a contradiction.
In Case (c), the number of cliques in with exactly vertices is equal to 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 be a non-geometric distance-regular graph with diameter , valency but without -claws. If is -claw-free, then is the icosahedron by [11]. Suppose that contains a -claw. If holds, then, by Lemma 5.2, satisfies (3), (4), or (5) of Theorem 5.1. If , then it follows by Proposition 5.15, Lemmas 5.16, 5.17, 5.19, and Proposition 5.20 that has the intersection array , i.e., 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).
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