Growth and the spectrum of the Laplacian of an infinite graph (Q1922260)

Let \(G\) be a locally finite, infinite graph and let \(\Delta\) be the combinatorial Laplacian. Let \(m\) and \(M\) be the infimum and the supremum of the essential spectrum of \(\Delta\); \(0\leq m\leq1\leq M\leq 2\). Let \(\mu(G)\) be the exponential growth constant of \(G\). The author shows that \[ m\leq{1-2e^{\mu/2}\over 1+e^\mu}\quad\text{and}\quad m\leq\Biggl({1-e^{\mu/2}\over 1+e^{\mu/2}}\Biggr)^2M. \] The first upper bound for \(m\) is a generalization of a result of Brooks for Riemannian manifolds. The author also derives several interesting corollaries of this result.