Spectra of infinite graphs with summable weight functions

TL;DR

This paper investigates the spectra of Laplacians on infinite weighted graphs with summable weights, introducing new invariants and inequalities to characterize spectral properties, especially for bipartite and complete graphs.

Contribution

It introduces a new combinatorial invariant k(G,m), extends Cheeger inequalities to summable weighted graphs, and provides detailed spectral analysis of infinite complete graphs.

Findings

Established a Cheeger inequality analogue for summable weighted graphs.

Introduced the invariant k(G,m) for spectral asymmetry and bipartiteness.

Analyzed spectra of infinite complete graphs in detail.

Abstract

In this paper we study spectra of Laplacians of infinite weighted graphs. Instead of the assumption of local finiteness we impose the condition of summability of the weight function. Such graphs correspond to reversible Markov chains with countable state spaces. We adopt the concept of the Cheeger constant to this setting and prove an analogue of the Cheeger inequality characterising the spectral gap. We also analyse the concept of the dual Cheeger constant originally introduced in \cite{B14}, which allows estimating the top of the spectrum. In this paper we also introduce a new combinatorial invariant, k$(G,m)$, which allows a complete characterisation of bipartite graphs and measures the asymmetry of the spectrum (the Hausdorff distance between the spectrum and its reflection at point $1\in \Bbb R$). We compare k$(G, m)$ to the Cheeger and the dual Cheeger constants. Finally, we analyse in full detail a class of infinite complete graphs and their spectra.