The Laplacian on rapidly branching trees (Q1918375)

Let \(K(r):= \sup\{K(x, \pi): d(p, x)\geq r\}\), where \(K(x, \pi)\) is the sectional curvature of a 2-plane in \(T_x M\) and \(p\) is the basepoint of a complete simply connected negatively curved Riemannian manifold \(M\). Let \(\text{Ess}(\Delta_M)\) be the essential spectrum of the Laplacian on M. Donnelly and Li proved: Theorem. If \(\lim_{r\to \infty} K(r)= -\infty\), then \(\text{Ess}(\Delta_M)\) is empty. The author establishes an analogous theorem for graphs. Let \(\alpha_\infty\) be the isoperimetric constant at infinity of a locally finite infinite graph \(G\) and let \(\Delta_G\) be the combinatorial Laplacian. Theorem. a) \(\alpha_\infty= 1\Leftrightarrow \text{Ess}(\Delta_G)= \{1\}\). b) If \(T\) is a rapidly branching tree, then \(\text{Ess}(\Delta_T)= \{1\}\). c) Let \(T\) be an infinite tree with \(m\geq 2\). If 1 is an eigenvalue of \(\Delta_T\), then it has infinite multiplicity. The author also gives necessary and sufficient conditions in terms of the branching numbers that 1 is an eigenvalue of \(\Delta_T\) if \(T\) is an infinite symmetric tree.