On the nullspace of arc-transitive graphs over finite fields

Abstract

Let A be the adjacency matrix of a graph Γ. The nullity of A (that is, the dimension of the nullspace of A), when viewed as a matrix over a field of prime characteristic p, is called the p-nullity of Γ. We present several families of arc-transitive graphs with arbitrarily large p-nullity. We also show that the p-nullity of a vertex-transitive graph of order a power of p is zero, provided that the valency of the graph is coprime to p.