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On cliques in edge-regular graphs Leonard Soicher Queen Mary University of London Modern Trends in Algebraic Graph Theory, Villanova, 2014 Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 1 / 10 The setup All graphs in this talk are

SLIDE 1

On cliques in edge-regular graphs

Leonard Soicher

Queen Mary University of London

Modern Trends in Algebraic Graph Theory, Villanova, 2014

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 1 / 10

SLIDE 2

The setup

All graphs in this talk are finite, undirected, and have no loops and no multiple edges.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 2 / 10

SLIDE 3

The setup

All graphs in this talk are finite, undirected, and have no loops and no multiple edges.

Definition

A graph Γ is edge-regular with parameters (v, k, λ) if Γ has exactly v vertices, is regular of valency k, and every pair of adjacent vertices have exactly λ common neighbours.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 2 / 10

SLIDE 4

The setup

All graphs in this talk are finite, undirected, and have no loops and no multiple edges.

Definition

A graph Γ is edge-regular with parameters (v, k, λ) if Γ has exactly v vertices, is regular of valency k, and every pair of adjacent vertices have exactly λ common neighbours.

Definition

A clique in a graph Γ is a set of pairwise adjacent vertices, an s-clique is a clique of size s, and a maximum clique of Γ is a clique of the largest size in Γ. The size of a maximum clique in Γ, its clique number, is denoted by ω(Γ).

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 2 / 10

SLIDE 5

Definition

A regular clique, or more specifically, an m-regular clique in a graph Γ is a non-empty clique S such that every vertex of Γ not in S is adjacent to exactly m vertices of S, for some constant m > 0.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 3 / 10

SLIDE 6

Definition

A regular clique, or more specifically, an m-regular clique in a graph Γ is a non-empty clique S such that every vertex of Γ not in S is adjacent to exactly m vertices of S, for some constant m > 0.

Definition

A quasiregular clique, or more specifically, an m-quasiregular clique in a graph Γ is a clique S of size at least 2, such that every vertex of Γ not in S is adjacent to exactly m or m + 1 vertices of S, for some constant m ≥ 0.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 3 / 10

SLIDE 7

I am interested in good upper bounds for the clique numbers of edge-regular graphs, given (only) their parameters.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 4 / 10

SLIDE 8

I am interested in good upper bounds for the clique numbers of edge-regular graphs, given (only) their parameters. I am also interested in properties of quasiregular cliques in edge-regular graphs.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 4 / 10

SLIDE 9

The clique adjacency polynomial

This is our main tool.

Definition

The clique adjacency polynomial of an edge-regular graph Γ with parameters (v, k, λ) is CΓ(x, y) = Cv,k,λ(x, y) := x(x + 1)(v − y) − 2xy(k − y + 1) + y(y − 1)(λ − y + 2).

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 5 / 10

SLIDE 10

The clique adjacency polynomial

This is our main tool.

Definition

The clique adjacency polynomial of an edge-regular graph Γ with parameters (v, k, λ) is CΓ(x, y) = Cv,k,λ(x, y) := x(x + 1)(v − y) − 2xy(k − y + 1) + y(y − 1)(λ − y + 2). This polynomial is a special case of the “block intersection polynomials” introduced by Cameron and S. (2007) and further studied by S. (2010). The theory of block intersection polynomials can be used to prove the following:

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 5 / 10

SLIDE 11

Theorem

Let Γ be an edge-regular graph with parameters (v, k, λ), let C(x, y) = Cv,k,λ(x, y), and suppose Γ has an s-clique S, with s ≥ 2. Then:

1 C(x, s) = s

i=0(i − x)(i − x − 1)ni, where ni is the number of

vertices of Γ not in S adjacent to exactly i vertices in S;

2 C(m, s) ≥ 0 for every integer m; 3 if m is a non-negative integer then C(m, s) = 0 if and only if S is

m-quasiregular, in which case the number of vertices outside S adjacent to exactly m vertices in S is C(m + 1, s)/2;

4 if m is a positive integer then C(m − 1, s) = C(m, s) = 0 if and only

if S is m-regular.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 6 / 10

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Generalisation of a result of Neumaier

We can apply the clique adjacency polynomial to prove the following theorem, which generalises a result of Neumaier (1981) on regular cliques in edge-regular graphs.

Theorem

Suppose Γ is an edge-regular graph, not complete multipartite, which has an m-quasiregular s-clique. Then for all edge-regular graphs ∆ with the same parameters (v, k, λ) as Γ:

1 ω(∆) ≤ s, so in particular, ω(Γ) = s; 2 all quasiregular cliques in ∆ are m-quasiregular cliques; 3 the quasiregular cliques in ∆ are precisely the cliques of size s

(although ∆ may have no cliques of size s).

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 7 / 10

SLIDE 13

Bounding the clique number of an edge-regular graph

A clique adjacency polynomial can be used to determine an upper bound on the clique number of an edge-regular graph Γ with given parameters (v, k, λ), as follows.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 8 / 10

SLIDE 14

Bounding the clique number of an edge-regular graph

A clique adjacency polynomial can be used to determine an upper bound on the clique number of an edge-regular graph Γ with given parameters (v, k, λ), as follows. Let C(x, y) = Cv,k,λ(x, y), and let b = bv,k,λ be the least positive integer such that C(m, b + 1) < 0 for some integer m. Then ω(Γ) ≤ b.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 8 / 10

SLIDE 15

Bounding the clique number of an edge-regular graph

A clique adjacency polynomial can be used to determine an upper bound on the clique number of an edge-regular graph Γ with given parameters (v, k, λ), as follows. Let C(x, y) = Cv,k,λ(x, y), and let b = bv,k,λ be the least positive integer such that C(m, b + 1) < 0 for some integer m. Then ω(Γ) ≤ b. Such a b always exists, and is easy to calculate.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 8 / 10

SLIDE 16

Bounding the clique number of an edge-regular graph

A clique adjacency polynomial can be used to determine an upper bound on the clique number of an edge-regular graph Γ with given parameters (v, k, λ), as follows. Let C(x, y) = Cv,k,λ(x, y), and let b = bv,k,λ be the least positive integer such that C(m, b + 1) < 0 for some integer m. Then ω(Γ) ≤ b. Such a b always exists, and is easy to calculate. I know of no case of a strongly regular graph with parameters (v, k, λ, µ) where the bound b = bv,k,λ is worse than the Delsarte-Hoffman bound, and some cases where the bound b is strictly better.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 8 / 10

SLIDE 17

Example

The parameters with smallest v for which the existence of a strongly regular graph is unknown are (v, k, λ, µ) = (65, 32, 15, 16). A strongly regular graph with these parameters would have least eigenvalue (−1 − √ 65)/2, and the Delsarte-Hoffman bound would be 8 = ⌊1 + 64/(1 + √ 65)⌋. However, we calculate that b65,32,15 = 7 (in particular, C65,32,15(3, 8) = −12), and so any edge-regular graph ∆ with parameters (65, 32, 15) has ω(∆) ≤ 7.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 9 / 10

SLIDE 18

Example

The parameters with smallest v for which the existence of a strongly regular graph is unknown are (v, k, λ, µ) = (65, 32, 15, 16). A strongly regular graph with these parameters would have least eigenvalue (−1 − √ 65)/2, and the Delsarte-Hoffman bound would be 8 = ⌊1 + 64/(1 + √ 65)⌋. However, we calculate that b65,32,15 = 7 (in particular, C65,32,15(3, 8) = −12), and so any edge-regular graph ∆ with parameters (65, 32, 15) has ω(∆) ≤ 7. Perhaps it would be fruitful to search for a strongly regular graph with parameters (65, 32, 15, 16) and containing a clique of size 7.

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 9 / 10

SLIDE 19

And finally

Some references:

1 P.J. Cameron and L.H. Soicher, Block intersection polynomials, Bull.

London Math. Soc. 39 (2007), 559–564.

2 A. Neumaier, Regular cliques in graphs and special 11

2-designs, in

Finite Geometries and Designs, Cameron et al., eds, London Math.

Soc. Lect. Note Ser. 49, Cambridge University Press, Cambridge,

1981, pp. 244–259.

3 L.H. Soicher, More on block intersection polynomials and new

applications to graphs and block designs, J. Combin. Theory Ser. A 117 (2010), 799–809.

4 L.H Soicher, The DESIGN package for GAP, Version 1.6, 2011,

http://designtheory.org/software/gap design/

5 L.H. Soicher, On cliques in edge-regular graphs, preprint,

http://www.maths.qmul.ac.uk/~leonard/ergcliques.pdf

Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 10 / 10