Abstract. The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.

A subset S of a group G ≤ Sym(n) is intersecting if for any pair of permutations π, σ ∈ S there is an i ∈ {1, 2, . . . , n} such that π(i) = σ (i). It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(n), Alt(n), and PGL(2, q) are exactly the cosets of the point-stabilizers. In this paper, we show how this approach can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transitive groups with degree no more than 20.

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A subset S of the alternating group on n points is intersecting if for any pair of permutations π, σ in S, there is an element i ∈ {1, . . . , n} such that π(i) = σ(i). We prove that if S is intersecting, then |S| ≤ (n−1)! 2 . Also, we prove that if n ≥ 5, then the only sets S that meet this bound are the cosets of the stabilizer of a point of {1, . . . , n}.

For any graph X with the adjacency matrix A, the transition matrix of the continuous-time quantum walk at time t is given by the matrix-valued function H X (t) = e itA . We say that there is perfect state transfer in X from the vertex u to the vertex v at time τ if |H X (τ ) u,v | = 1. It is an important problem to determine whether perfect state transfers can happen on a given family of graphs. In this paper we characterize all the graphs in the Johnson scheme which have this property. Indeed, we show that the Kneser graph K(2k, k) is the only class in the scheme which admits perfect state transfers. We also show that, under some conditions, some of the unions of the graphs in the Johnson scheme admit perfect state transfer.