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.

The authors’ thanks are due to Professor Gerhard Pazderski (Wilhelm-Pieck-Universität Rostock) for valuable discussions and comments.