On the bottom of the spectrum of the Laplacian on graphs (Q2779206)

The author studies the combinatorial Laplacian on a graph \(\Gamma\) whose edges have variable length. Let \(\lambda(\Gamma)\) be the bottom of the spectrum of the Laplacian on \(\Gamma\) and let \(\ell_0\) be the inf of the edge lengths. The author defines an isoperimetric constant \(\alpha\) so that the following estimate holds: \({{\alpha(\Gamma)}\over{\ell}} \geq\lambda(\Gamma)\geq{1\over 2}\alpha^2(\Gamma)\). This generalizes previously known results for graphs whose edges have constant lengths to the more general setting.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].