On the clique number of a strongly regular graph

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.


Introduction
The clique number ω(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 ω(Γ) ⌊1 − k/s⌋; 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 = p k be a power of a prime p congruent to 1 mod 4. A Paley graph has vertex set equal to the finite field F q , 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, Blokhuis [4] showed that ω(Γ) = √ q; this 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 certain such q, showed that ω(Γ) ⌊ √ q − 1⌋. 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 parameters 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 Section 4), 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.
The paper is organised as follows. In Section 2 we state our main results and in Section 3 we state some standard identities that we will use in our proofs. Section 4 contains the proofs of our main results. In Section 5 we examine the strength of the clique adjacency bound and in Section 6 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.

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 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 certain nonnegativity and divisibility constraints given by Brouwer [6,VII.11.5]).
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 [14,Section 10.2]). 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 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 ∈ R is defined as frac (a) := a − ⌊a⌋. We are now ready to state our main results.
Theorem 2.1. Let Γ be a type-I strongly regular graph with v vertices. Suppose that Let P denote the set of all primes p of the form p = 1 + 4g for some g ∈ N. Then the sequence ( √ p/2) p∈P is uniformly distributed modulo 1 (see Balog [2,Theorem 1]). Therefore, since Paley graphs on p vertices exist for all p ∈ P, Theorem 2.1 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 Theorem 2.1.
A graph is called co-connected if its complement is connected. We have the following: Theorem 2.4. Let Γ be a co-connected type-II strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s. Suppose that Then ω(Γ) ⌊−k/s⌋.
Proof. Follows from Theorem 4.1 together with Corollary 4.9 below.
Remark 2.5. 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 Theorem 2.4. By the remark following Corollary 4.9, it follows that Theorem 2.4 can be applied to about 1/4 of the complementary pairs of type-II strongly regular graphs on Brouwer's list.
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 Theorem 2.4; in fact, in this case, the conclusion of Theorem 2.4 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 Theorem 2.4 and whose corresponding graphs are not yet known to exist (or not exist).

Parameters of strongly regular graphs
Here we state some well-known properties of strongly regular graphs and their parameters. The first two propositions are standard (see Brouwer and Haemers [8,Chapter 9] or Cameron and Van Lint [9, Chapter 2]). Proposition 3.1. Let Γ be a strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s. Then Proposition 3.2. Let Γ be a type-I strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r > s. Then The next proposition is a key observation.
Let Γ be a strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s.
Proof. If Γ is type I then, by Proposition 3.2, we have k − 2µ = 0. If Γ is type II then, by Proposition 3.1, we have k/s − µ/s = −r and r is an integer.
Next, the complement Γ of a strongly regular graph Γ is also a strongly regular graph. This is again a standard result (see Cameron and Van Lint [9, Chapter 2]).
Proposition 3.4. Let Γ be a connected and co-connected strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r > s. Then Γ is strongly regular Finally we state some straightforward bounds for the parameters of strongly regular graphs.

The clique adjacency polynomial
Now we define our main tool, the clique adjacency polynomial. Given an edgeregular graph Γ with parameters (v, k, λ), define the clique adjacency polynomial C Γ (x, y) as The utility of the clique adjacency polynomial follows from [16, Theorem 1.1] (see also [17,Theorem 3.1]), giving: Theorem 4.1. Let Γ be an edge-regular graph with parameters (v, k, λ). Suppose that Γ has a clique of size c 2. Then C Γ (b, c) 0 for all integers b.
As discussed in [16] and [17], Theorem 4.1 provides a way of bounding the clique number of an edge-regular graph using only its parameters. Indeed, by Theorem 4.1, for an edge-regular graph Γ and some integer c 2, if there exists an integer b such that C Γ (b, c) < 0 then ω(Γ) c−1. Hence we define the clique adjacency bound (CAB) to be the least integer c 2 such that C Γ (b, c + 1) < 0 for some b ∈ Z; note that such a c always exists.
We will show that, for a k-regular strongly regular graph Γ, the clique adjacency bound gives ω(Γ) ⌊1 − k/s⌋. That is, the clique adjacency bound is always at least as good as the Delsarte bound when applied to strongly regular graphs. This follows from Theorem 4.1 together with Theorem 4.2 below. More interestingly, we will also show that the clique adjacency bound does better than the Delsarte bound for infinitely many feasible parameter tuples for strongly regular graphs. In this section we consider the univariate polynomial C Γ (f (t), g(t)) in the variable t, where f (t) and g(t) are linear polynomials in t. The main idea is to choose the linear polynomials f and g such that there exists t ∈ R such that C Γ (f (t), g(t)) < 0, f (t) ∈ Z, and g(t) is an integer at least 2. We begin by stating one of the main results of this paper. Theorem 4.2. Let Γ be a strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s. Then, Proof. Follows from Corollary 4.5 and Corollary 4.8 below.
Observe that, together with Theorem 4.1, Theorem 4.2 shows that the clique adjacency bound always does as well as the Delsarte bound for strongly regular graphs.
Now we can state our first polynomial identity, which shows that the clique adjacency polynomial is negative at a certain point.
Let Γ be a connected strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r > s. Then Proof. The equality follows from direct calculation (see Appendix A), using the equalities in Proposition 3.1. The right-hand side is negative since s < 0 and r 0.
Let Γ be a strongly regular graph with parameters (v, k, λ, µ) such that both µ/s and k/s are integers. Then by Lemma 4.3, together with Theorem 4.1, we recover the Delsarte bound, i.e., ω(Γ) ⌊1 − k/s⌋. It remains for us to deal with the situation when k/s and µ/s are not integers. In the remainder of this section, motivated by Lemma 4.3, we consider integral points (x, y) ∈ Z 2 close to (−µ/s, 2 − k/s) such that C Γ (x, y) is negative. We deal with the type I and type II cases separately.

4.1.
Type-I strongly regular graphs. Let Γ be a type-I strongly regular graph (or conference graph) with v vertices. By Proposition 3.2 we have −µ/s = r and −k/s = 2r. Therefore, we consider integral points (x, y) close to (r, 2 + 2r) at which to evaluate the clique adjacency polynomial. In view of Proposition 3.3, we evaluate C Γ (x, y) at points of the form (r − t, a + 2r − 2t) for some a ∈ N, thinking of t as the fractional part of r.
Lemma 4.4. Let Γ be a type-I strongly regular graph with v vertices and eigenvalues k > r > s. Then Proof. The equalities follow from direct calculation (see Appendix A), applying Proposition 3.1 and the definition of a type-I strongly regular graph.
The right-hand side of Equation (2) is a cubic polynomial in the indeterminate t with positive leading coefficient. Furthermore, since for a type-I strongly regular graph we have s = −( √ v + 1)/2, we observe that the smallest zero of the right-hand side of Equation (2) is equal to 3/4+( √ v− v + 5/4)/2. Hence C Γ (r−t, 2+2r−2t) is negative for t < 3/4 + ( √ v − v + 5/4)/2. We use this observation in the next result, which can be used with Theorem 4.1 to obtain the Delsarte bound for conference graphs.
The next corollary follows in a similar fashion.

4.2.
Type-II strongly regular graphs. Let Γ be a type-II strongly regular graph with parameters (v, k, λ, µ). Again, in view of Proposition 3.3, we evaluate C Γ (x, y) at points of the form (−µ/s − t, a − k/s − t) for some a ∈ Z, thinking of t as the fractional part of −µ/s. Lemma 4.7. Let Γ be a strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s. Then Moreover, if Γ is co-connected then these polynomials have positive leading coefficients.
Proof. The equalities follow from direct calculation (see Appendix A), using the equalities in Proposition 3.1. By Proposition 3.5 if Γ is co-connected then the polynomials have positive leading coefficients.
Let t = frac (−µ/s). Then, using Proposition 3.3 and Equation (3), we have Suppose first that Γ is co-connected. The right-hand side of Equation (3) is negative on the open interval (η, 1), where η = (2s − r)(r + 1)/(v − 2k + λ) is negative. Hence the corollary holds for Γ. On the other hand, for complete multipartite graphs we have t = 0, in which case the right-hand side of Equation (3) is negative.
The next corollary follows similarly, using the fact that the right-hand side of Equation (4)  Corollary 4.9. Let Γ be a co-connected type-II strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s. Suppose that Then C Γ (⌊−µ/s⌋, ⌊1 − k/s⌋) < 0.
Remark 4.10. We remark that if a type-II strongly regular graph satisfies the hypothesis of Corollary 4.9 then its complement cannot also satisfy the hypothesis. Indeed, suppose that Γ satisfies the hypothesis of Corollary 4.9. Since frac (−k/s) > 0 we have that s = −1 and hence Γ is connected. Then, using Proposition 3.4, we see that the complement of Γ also satisfies the hypothesis of Corollary 4.9 if 0 < frac ((v − k − 1)/(r + 1)) < 1 − (s 2 + s)/µ.

How sharp is the clique adjacency bound?
In this section we show that the clique adjacency bound is sharp for strongly regular graphs in certain instances. We also comment on the sharpness of the clique adjacency bound for general strongly regular graphs.
Theorem 5.1. Let Γ be a strongly regular graph with parameters (v, k, λ, µ) and eigenvalues k > r s. Suppose that λ + 1 −k/s. Then the clique adjacency bound is equal to λ + 2.
Since s < −1, it follows that r + 1 > −k/s. Multiplying this inequality by −s gives −s(r + 1) > k. Since both s and r are integers, we have −s(r + 1) k + 1 Now by rearranging and substituting µ = k + rs, we obtain the inequality 1 + (µ + 1)/s 0 as required. Proof. Let c ∈ {2, . . . , λ + 2} and let b be an integer. If b 0, then from the definition of the clique adjacency polynomial C Γ (x, y), we see that C Γ (b, c) 0, so we now assume that b is positive.
A calculation (see Appendix A) shows that This quantity is nonnegative since b and λ + 2 − c are nonnegative integers, the product of two consecutive integers is nonnegative, and k − λ− 1 is also nonnegative by Proposition 3.5. Hence as required. Now we prove Theorem 5.1.
Proof of Theorem 5.1. Firstly, if Γ is disconnected then Γ is the disjoint union of complete graphs and hence contains cliques of size λ + 2. Therefore the clique adjacency bound is at least λ + 2. Now we assume that Γ is connected. By Lemma 5.3, the clique adjacency bound is less than λ + 2 only if there exists some integer b such that C Γ (b, λ + 2) is less than zero. To ease notation set f (x) := C Γ (x, λ + 2). Hence It suffices to show that there does not exist any integer b such that f (b) < 0.
Observe that the polynomial f (x) is a quadratic polynomial in the variable x. Furthermore, the leading coefficient of f (x) is v − λ − 2 0, and f (0) = 0. Let ξ be the other zero of f (x). Now, f (x) is negative if and only if x is between 0 and ξ. Hence, if f (−1) and f (1) are both nonnegative then there are no integers b such that f (b) < 0. As in the proof of the previous result f (−1) is nonnegative. Therefore Lemma 5.2 completes the proof for type-II strongly regular graphs.
The inequality λ + 1 −k/s only holds for type-I strongly regular graphs on 5 vertices or 9 vertices (where we have equality). One can explicitly compute the clique adjacency bound for these two cases: the unique (5, 2, 0, 1)-strongly regular graph and the unique (9, 4, 1, 2)-strongly regular graph. For each of these graphs the clique adjacency bound is equal to λ + 2. Now we give a couple of remarks about Theorem 5.1.
Remark 5.4. For strongly regular graphs with λ 1, it is easy to see that the clique number is λ + 2. By Theorem 5.1, the clique adjacency bound is equal to the clique number for such graphs. Let Γ be a strongly regular graph with parameters (v, k, λ, µ). By Proposition 3.1, we see that k = −s(r + 1) − r + λ. Therefore, for strongly regular graphs with λ = 2 and r 2, we have λ + 1 = 3 −k/s, and so Theorem 5.1 applies to such graphs.
Remark 5.5. We conjecture that if the clique adjacency bound is less than −k/s then λ + 1 −k/s. We have verified this conjecture for all feasible parameter tuples for strongly regular graphs on up to 1300 vertices, making use of Brouwer's website [7].
In Table 1, we list all the feasible parameter tuples for strongly regular graphs on at most 150 vertices to which we can apply either Theorem 2.1 or Theorem 2.4. In other words, Table 1 displays the feasible parameters for strongly regular graphs on at most 150 vertices for which the clique adjacency bound is strictly less than the Delsarte bound. In the column labelled 'Exists', if there exists a strongly regular graph with the appropriate parameters then we put '+', or '!' if the graph is known to be unique; otherwise, if the existence is unknown, we put '?'. In the final column of Table 1, we put 'Y' (resp. 'N') if there exists (resp. does not exist) a strongly regular graph with the corresponding parameters that has clique number equal to the clique adjacency bound, otherwise we put a '?' if such existence is unknown. We refer to Brouwer's website [7] for details on the existence of strongly regular graphs with given parameters.
For the parameter tuples in Table 1, the Delsarte bound is equal to the clique adjacency bound plus 1. As an example of a parameter tuple for which the clique adjacency bound differs from the Delsarte bound by 2, we have (378, 52, 1, 8) for which there exists a corresponding graph [11]. For this graph the Delsarte bound is 5, but the clique adjacency bound is 3.
Feasible parameters for which there does not exist a corresponding strongly regular graph whose clique number is equal to the clique adjacency bound include (16,10,6,6) and (27,16,10,8). However, we ask the following question. Do there exist strongly regular graphs with parameters (v, k, λ, µ), with k < v/2, such that every strongly regular graph having those parameters has clique number less than the clique adjacency bound?

Hoffman bound vs Delsarte bound vs clique adjacency bound
Let Γ be a connected non-complete regular graph with v vertices, valency k, and second largest eigenvalue r < k. Then the complement Γ of Γ is a regular graph with valency k = v − k − 1 and least eigenvalue s = −r − 1 < 0. We may obtain a bound for the clique number of Γ by applying the Hoffman bound (also called the ratio bound) [13,Theorem 2.4.1] on the size of a largest independent set (coclique) of Γ. This gives If Γ is strongly regular, then it is known (and follows from the relations of Proposition 3.1) that the Delsarte bound for ω(Γ) is the same as that given by the Hoffman bound above. Now the Delsarte bound applies not only to strongly regular graphs, but also to the graphs {Γ 1 , . . . , Γ d } of the relations (other than equality) of any d-class symmetric association scheme (see [13,Corollary 3.7.2]). Thus, if Γ is such a graph, having valency k and least eigenvalue s, then ω(Γ) ⌊1 − k/s⌋.
Here is an interesting illustrative example. Let ∆ be the edge graph (or line graph) of the incidence graph of the projective plane of order 2. Then ∆ is the unique distance-regular graph with intersection array {4, 2, 2; 1, 1, 2}. Now let ∆ 3 be the graph on the vertices of ∆, with two vertices joined by an edge if and only if they have distance 3 in ∆. Then ∆ 3 is the graph of a relation in the usual symmetric association scheme associated with a distance-regular graph, where two vertices are in relation i precisely when they are at distance i in the distance-regular graph. The graph ∆ 3 has diameter 2 and is edge-regular (but not strongly regular) with parameters (v, k, λ) = (21, 8, 3). The clique adjacency bound for ∆ 3 is 4.
The least eigenvalue of ∆ 3 is − √ 8, and the Delsarte bound gives 3, and indeed, ω(∆ 3 ) = 3. However, the complement of ∆ 3 has least eigenvalue −1 − √ 8, and the Hoffman bound for independent sets in the complement of ∆ 3 gives 5. Thus, for ∆ 3 , the Delsarte bound is better than the clique adjacency bound which is better than that obtained from the Hoffman bound. However, the three bounds are for different classes of graphs. For example, there may well be an edge-regular graph with parameters (21, 8, 3) and clique number 4. It would be interesting to find one.
We conjecture that if Γ is any connected non-complete edge-regular graph, then the clique adjacency bound for ω(Γ) is at most that obtained from the Hoffman bound for Γ.

Appendix A. Algebraic computational verification of identities
In this appendix we present the algebraic computations in Maple [3] that were used to verify certain identities employed in this paper. These identities were also checked independently using Magma [5].
We start up Maple (version 18) and assign to C the clique adjacency polynomial.