The spectral radius of infinite graphs

Recently some important results have been proved showing that the gap between the largest eigenvalue A: of a finite regular graph of valency k and its second eigenvalue is related to expansion properties of the graph [1]. In this paper we investigate infinite graphs and show that in this case the expansion properties are related to the spectral radius of the graph. First we introduce necessary notions for the spectrum of an infinite graph following the definitions of [7]. For an infinite graph F with vertex set V and finitely bounded valency, the adjacency operator A is well-defined on lV) and is bounded and self-adjoint. The spectrum of F is the approximate point spectrum of A in the space lV); that is A e Spec, 4 if and only if there is a sequence of unit vectors xn such