Automorphism group and spectrum of a graph

Abstract

It is well known that there are close relations between the automorphism group aut G and the spectrum of a (finite, directed or undirected) graph G which can be investigated by the general methods of representation theory, or by more direct methods. aut G can be represented as the group g (A) of all permutation matrices P which commute with the adjacency matrix A of G. Therefore, if x is any eigenvector of A belonging to the eigenvalue λ, then so is Px for each P ∈ g (A). If, for some P ∈ g (A), x and Px prove linearly independent then λ must have a multiplicity m > 1. So, if aut G is rich enough, the occurrence of a simple (non-trivial) eigenvalue is an “exception”. In this paper it is assumed that aut G is transitive, and it is investigated under what conditions simple eigenvalues can occur. In particular, a sharp upper bound for the number of simple eigenvalues (in terms of aut G) is given, and from the intermediate results some conclusions are drawn.