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On the Clique Number of a Strongly Regular Graph

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On the Clique Number of a Strongly Regular Graph On the clique number of a strongly regular graph Gary R. W. Greaves∗ Leonard H. Soicher Division of Mathematical Sciences, School of Mathematical Sciences, School of Physical and Mathematical Sciences, Queen Mary University of London, Nanyang Technological University, Mile End Road, London E1 4NS, UK 21 Nanyang Link, Singapore 637371, Singapore [email protected] [email protected] Submitted: May 18, 2018; Accepted: Sep 23, 2018; Published: Oct 19, 2018 c The authors. Released under the CC BY-ND license (International 4.0). Abstract We determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. These bounds improve on the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs, including infinitely many parameter tuples that correspond to Paley graphs. Mathematics Subject Classifications: 05C69, 05E30 1 Introduction The clique number !(Γ) of a graph Γ is defined to be the cardinality of a clique of maximum size in Γ. For a k-regular strongly regular graph with smallest eigenvalue s < 0, Delsarte [12, Section 3.3.2] proved that !(Γ) 6 b1 − k=sc; we refer to this bound as the Delsarte bound. Therefore, since one can write s in terms of the parameters of Γ, one can determine the Delsarte bound knowing only the parameters (v; k; λ, µ) of Γ. In this paper we determine new upper bounds for the clique numbers of strongly regular graphs in terms of their parameters. Our bounds improve on the Delsarte bound infinitely often. Let q = pk be a power of a prime p congruent to 1 mod 4. A Paley graph has vertex set equal to the finite field Fq, and two vertices a and b are adjacent if and only if a − b is a nonzero square. For a Paley graph Γ on q vertices with k even, it is well-known1 ∗The first author was supported by JSPS KAKENHI; grant number: 26·03903 p 1The elements of the subfield of size q form a clique B. Blokhuis [4] showed that every clique of size p q is obtained from a linear transformation of B. the electronic journal of combinatorics 25(4) (2018), #P4.15 1 p that !(Γ) = q, which corresponds to equality in the Delsarte bound. Bachoc et al. [1] recently considered the case when Γ is a Paley graph on q vertices with k odd and, for p certain such q, showed that !(Γ) 6 b q − 1c. This corresponds to an improvement to the Delsarte bound for these Paley graphs. Here, working much more generally, given a strongly regular graph Γ with parameters (v; k; λ, µ), we provide inequalities in terms of the parameters of Γ that, when satisfied, guarantee that the clique number of Γ is strictly less than the Delsarte bound. We show that these inequalities are satisfied by infinitely many feasible parameter tuples for strongly regular graphs and, in particular, are satisfied by infinitely many parameter tuples that correspond to Paley graphs. Our inequalities are obtained using what we call the \clique adjacency bound" (see Section4), a bound defined by the second author [17]. We also show that the clique adjacency bound is always at most the Delsarte bound when applied to strongly regular graphs. We have made use of computer algebra assistance (using Maple Version 18 [3] and its Groebner package) to perform the direct calculations needed to verify certain equalities used in our proofs. We remark that the total CPU time for all this verification on a desktop Linux PC was only about 0.16 seconds, and the total store used by Maple was 2.4MB. The paper is organised as follows. In Section2 we state our main results and in Section3 we state some standard identities that we will use in our proofs. Section4 contains the proofs of our main results. In Section5 we examine the strength of the clique adjacency bound and in Section6 we provide an illustrative example comparing certain bounds for the clique number of an edge-regular graph that is not necessarily strongly regular. Finally, we give an appendix in which we describe our symbolic computations. 2 Definitions and main results A non-empty k-regular graph on v vertices is called edge-regular if there exists a constant λ such that every pair of adjacent vertices has precisely λ common neighbours. The triple (v; k; λ) is called the parameter tuple of such a graph. A strongly regular graph Γ with parameter tuple (v; k; λ, µ) is defined to be a non-complete edge-regular graph with parameter tuple (v; k; λ) such that every pair of distinct non-adjacent vertices has precisely µ common neighbours. We refer to the elements of the parameter tuple as the parameters of Γ. We call the parameter tuple of a strongly regular graph feasible if its elements satisfy the nonnegativity and divisibility constraints given by Brouwer [6, VII.11.5]); see the results of Section3. Let Γ be a strongly regular graph with parameters (v; k; λ, µ). It is well-known that Γ has at most three distinct eigenvalues, and moreover, the eigenvalues can be written in terms of the parameters of Γ (see Cameron and Van Lint [9]). In what follows we denote the eigenvalues of Γ as k > r > s. Strongly regular graphs whose parameters satisfy k = (v − 1)=2, λ = (v − 5)=4, and µ = (v − 1)=4 are called type I or conference graphs. Strongly regular graphs all of whose eigenvalues are integers are called type II. Every strongly regular graph is either the electronic journal of combinatorics 25(4) (2018), #P4.15 2 type I, type II, or both type I and type II (see Cameron and Van Lint [9, Chapter 2]). The fractional part of a real number a 2 R is defined as frac (a) := a − bac. We are now ready to state our main results. Theorem 1. Let Γ be a type-I strongly regular graph with v vertices. Suppose that p p 0 < frac v=2 < 1=4 − (pv + 5=4 − v)=2: p Then !(Γ) 6 b v − 1c. Proof. Follows from Theorem 11 together with Corollary 16 below. p p Remark 2. For a prime p satisfying 0 < frac p=2 < 1=4−(pp + 5=4− p)=2, we have p p p p that b pc = 2b p=2c is even. Furthermore, for n := b pc, since b pc > pp + 5=4−1=2, 2 p we have n + n − 1 > p. Hence, if Γ is a Paley graph on p vertices then !(Γ) 6 b p − 1c by Bachoc et al. [1, Theorem 2.1 (i)] (see [1, Remark 2.5]). Therefore, for type-I strongly p regular graphs with p (a prime) vertices, satisfying 0 < frac p=2 < 1=4 − (pp + 5=4 − p p)=2, Theorem1 is a generalisation of Bachoc et al. [1, Theorem 2.1 (i)]. Remark 3. Let g be a positive integer. Then (1 + 4g; 2g; g − 1; g) is a feasible pa- rameter tuple forp a type-I strongly regular graph on v = 1 + 4g vertices. Observe that (pv + 5=4 − v)=2 tends to 0 as v tends to infinity. Using Fej´er'stheorem (see Kuipersp and Niederreiter [15, page 13]), it is straightforward to show that the sequence ( 1 + 4g=2)g2N is uniformly distributed modulo 1. Therefore we can apply Theorem1 to about a quarter of all feasible parameter tuples for type-I strongly regular graphs. Let P denote the set of all primes p of the form p = 1 + 4g for some g 2 . Then p N the sequence ( p=2)p2P is uniformly distributed modulo 1 (see Balog [2, Theorem 1]). Therefore, since Paley graphs on p vertices exist for all p 2 P, Theorem1 is applicable to infinitely many strongly regular graphs. Note that the example in [17] with parameters (65; 32; 15; 16) is an example of a (potential) graph satisfying the hypothesis of Theorem1. A graph is called co-connected if its complement is connected. Note that strongly regular graphs that are not co-connected are complete multipartite. We have the following: Theorem 4. Let Γ be a co-connected type-II strongly regular graph with parameters (v; k; λ, µ) and eigenvalues k > r > s. Suppose that 0 < frac (−k=s) < 1 − (r2 + r)=(v − 2k + λ): Then !(Γ) 6 b−k=sc. Proof. Follows from Theorem 11 together with Corollary 19 below. Remark 5. Currently Brouwer [7] lists the feasible parameter tuples for connected and co-connected strongly regular graphs on up to 1300 vertices. Of these, about 1=8 of the parameter tuples of type-II strongly regular graphs satisfy the hypothesis of Theorem4. By the remark following Corollary 19, it follows that Theorem4 can be applied to about 1=4 of the complementary pairs of type-II strongly regular graphs on Brouwer's list. the electronic journal of combinatorics 25(4) (2018), #P4.15 3 Note that the example in [17] of a strongly regular graph with parameter tuple (144; 39; 6; 12) is an example of a graph satisfying the hypothesis of Theorem4; in fact, in this case, the conclusion of Theorem4 is satisfied with equality. The parameter tuple (88; 27; 6; 9) is the first parameter tuple in Brouwer's list to which we can apply Theorem4 and whose corresponding graphs are not yet known to exist (or not exist). 3 Parameters of strongly regular graphs Here we state some well-known properties of strongly regular graphs and their parameters.
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