Eigenvalues and automorphisms of a graph

Abstract

Let G be the automorphism group of a graph Γ and let λ be an eigenvalue of the adjacency matrix of Γ. In this article, (i) we derive an upper bound for rank(G), (ii) if G is vertex transitive, we derive an upper bound for the extension degree of ℚ(λ) over ℚ, (iii) we study automorphism groups of graphs without multiple eigenvalues, (iv) we study spectra of quotient graphs associated with orbit partitions.