The second eigenvalue of some normal Cayley graphs of highly transitive groups

Abstract

Let G be a finite group acting transitively on [n] = {1, 2,..., n}, and let Γ = Cay(G, T) be a Cayley graph of G. The graph Γ is called normal if T is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph Γ in terms of the second eigenvalues of certain subgraphs of Γ. Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of Sn, and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of Sn with maxTεT |supp(τ)| ≤ 5, where supp(τ) is the set of points in [n] non-fixed by τ.