Many of the graphs we consider in this book admit a large group of automorphisms. For example, the symmetric group Sym(n) acts as a group of automorphisms of the Johnson graph J (n, k). In some situations it is possible to use information about the group to derive information about the eigenvalues and eigenvectors of the graph, and this can allow us to derive theorems of EKR type. The group theoretic information needed comes from representation theory, and this chapter provides a reasonably self-contained introduction to this subject.

Representations

A representation Φ of a group G over the field F is a homomorphism from G into the group of invertible linear maps of some vector space V over F. For our purposes G is finite and V is a finite-dimensional vector space over C. The dimension of V is called the dimension of the representation. If Φ is a representation of G, we will denote the image of an element v of V under the action of an element g of G by Φ (g) v, or simply by gv when the choice of representation is irrelevant. As a representation corresponds to a homomorphism of the group algebra F[G] into End(V), it follows that V is a G-module. Hence we can express our thoughts on representations using the language of modules; this is often more convenient for developing the theory, but can be less useful when we have to do calculations.

We consider some examples. The first and simplest is the trivial representation. This representation maps each element of G to the identity map on a 1-dimensional vector space. We will often denote the trivial representation of G by 1G.

For our next example, let Gbe a permutation group on a set Ωand let V = F Ω. We can identify the points of Ω with the standard basis of V. Each element g of G then determines a permutation of this basis, and therefore it determines an endomorphism Φ (g) of V. We say Φ is the permutation representation of G. (The term permutation representation is also used to denote a homomorphism from a group in the symmetric group; we will provide a warning if there is a risk of confusion.)

We have seen many EKR-type theorems for objects other than sets. In Chapter 9 we gave a version of the EKR Theorem for vector spaces; in Chapter 10 we proved an EKR-type theorem for integer sequences. In this chapter we consider how an EKR-type theorem can be established for permutations.

To start, we recall from Section 7.5 our definition of intersection for permutations: two permutations π, σ? Sym(n) are said to be intersectingif π(i) = σ(i) for some i∈ ﹛1 , …, n﹜. Such permutations are also said to agreeon the point i. For an integer t≥ 1, a pair of permutations from Sym(n) are said to be t-intersectingif they agree on at least tpoints from ﹛1 , …, n﹜. A set of permutations is called intersecting (or t-intersecting) if any two permutations in the set are intersecting (or t-intersecting).

An example of a set of intersecting permutations is the set of all permutations in Sym(n) that fix some point i(this is the stabilizerof the point i). The stabilizer of a point is a set of intersecting permutations of size (n- 1)!. Similarly, the pointwise stabilizer of any tdistinct elements is a set of t-intersecting permutations with size (n- t)!. Further, any coset of these groups is also a set of t-intersecting permutations of size (n- t)!.

One generalization (perhaps the most natural) of the EKR Theorem for permutations is the assertion that the largest set of intersecting permutations has size (n - 1)!, and the only sets that meet this bound are the cosets of point stabilizers. Cameron and Ku [40] and Larose and Malvenuto [113] independently proved this result in 2006. Since then other proofs of this result have appeared (see [168, Section 2] for a particularly simple proof).

In this chapter we present a proof that uses the ratio bound for cocliques, and an analysis of the corresponding eigenspaces. This is the proof given in [85]. We will consider extending this result to sets of t-intersecting permutations.We will also look at the largest subsets of intersecting permutations from subgroups of the symmetric group.

A critical family of association schemes arise from distance-regular graphs. A graph is distance regular if for any two vertices u and v at distance k in the graph, the number of vertices w that are at distance i from u and j from v depends only on k, and not on the vertices u and v.

Suppose X is a graph with diameter d. The ith distance graph Xi of X has the same vertex set as X, and two vertices are adjacent in Xi if and only if they are distance i in X. For any graph X, the adjacency matrices of the distance graphs of X are called the distance matrices. The distance matrices for any graph are linearly independent and sum to J − I. It has been left as an exercise to show that if X is a distance-regular graph, then the distance matrices form an association scheme. Such an association scheme is called metric; these are discussed in detail in Section 4.1.

Let X be a graph and S a subset of the vertices of X. Denote by Si the set of vertices of X at distance i from S. The distance partition of X relative to S is the partition δS = { S1, …, Sr } of V (X). We say that S is a completely regular subset of X if δS is an equitable partition. If X is a distance-regular graph, then for any x ∈ V (X) the set S = { x} is completely regular. Moreover, the quotient graphs of X with respect to each δ{ x} are isomorphic. For any distance partition δS = { S1, …, Sr } of X, vertices in Si can only be adjacent to vertices in Si−1, Si and Si+1.

A graph on v vertices is a strongly regular graph with parameters (v, k; a, c) if:

1. (a) it is k-regular;

2. (b) each pair of adjacent vertices in the graph have exactly a common neighbors;

3. (c) each pair of distinct nonadjacent vertices in the graph have exactly c common neighbors.

If X is a strongly regular graph, then its complement is also a strongly regular graph. A strongly regular graph X is primitive if both X and its complement are connected. If X is not primitive, we call it imprimitive. The imprimitive strongly regular graphs are exactly the disjoint unions of complete graphs and their complements, the complete multipartite graph. It is not difficult to show that a strongly regular graph is primitive if and only if 0 < c < k. It is customary to declare that the complete and empty graphs are not strongly regular, and we follow this custom.

Two examples of strongly regular graphs are the line graphs of complete graphs and the line graphs of complete bipartite graphs. We will meet other large classes of examples as we go through this chapter – most of which arise from well-known combinatorial objects. In this chapter we provide detailed information about the cliques and cocliques for many of these graphs. We see that many interesting objects from design theory and finite geometry occur encoded as cliques and cocliques in these graphs, and this leads to interesting variants of the EKR Theorem.

An association scheme

As stated in the previous chapter, association schemes with two classes are equivalent to strongly regular graphs. Throughout this section we denote the adjacency matrix of a graph X by A, rather than A (X), and the adjacency matrix of the complement of X by Ā, rather than.

Lemma. Let X be a graph. Then X is strongly regular if and only if A = { I,A,Ā} is an association scheme.

Proof. Assume that X is a strongly regular graph with parameters (v, k; a, c). It is clear that the set of matrices A = { I,A, Ā} satisfies properties (a) through (c) in the definition of an association scheme given in Section 3.1.We only need to confirm that the product of any two matrices in A is a linear combination of other matrices in A (in doing this, we also show that the matrices commute).