A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs

A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph G with diameter k and vertex set V , the smallest nontrivial eigenvalue λ1 of the normalized Laplacian L satisfies λ1 6 1− σ ( 1− c k ) for some constant c where σ = 2 ∑ v dv √ dv − 1/ ∑ v d 2 v and dv denotes the degree of the vertex v. We consider weak Ramanujan graphs defined as graphs satisfying λ1 > 1 − σ. We examine the vertex expansion and edge expansion of weak Ramanujan graphs and then use the expansion properties among other methods to derive the above Alon-Boppana bound.