Boundary area growth and the spectrum of discrete Laplacian (Q1403394)

For an infinite graph \(G\), let \(\lambda_0^{\text{ess}}(G)\) denote the bottom of the essential spectrum of the discrete Laplacian of \(G\). The main result of the paper gives an upper bound of \(\lambda_0^{\text{ess}}(G)\) by a new geometric quantity \(\mu\), called the boundary area growth of the graph. Precisely: \[ \lambda_0^{\text{ess}}(G)\leq 1- (\cosh (\mu/2))^{-1}. \] A Riemannian analogue of this inequality had previously been obtained by the author in [Koday Math. J. 24, No. 1, 42--47 (2001; Zbl 0987.58016)]. An upper bound of \(\lambda_0^{\text{ess}}(G)\) by the volume growth was obtained by \textit{K. Fujiwara} [Tohoku Math J. 48, No.2, 293--302 (1996; Zbl 0857.05070)], after the corresponding, original estimate by \textit{R. Brooks} [Math. Z. 178, 501--508 (1981; Zbl 0458.58024)] in the Riemannian case. The present result refines Fujiwara's estimate.