On adjacency operators of locally finite graphs

Abstract

A graph $\Gamma$ is called locally finite if, for each vertex $v$ of $\Gamma$, the set $\Gamma (v)$ of all neighbors of $v$ in $\Gamma$ is finite. For any locally finite graph $\Gamma$ with vertex set $V(\Gamma )$ and for any field $F$, let $F^{V(\Gamma )}$ be the vector space over $F$ of all functions $V(\Gamma )\to F$ (with natural componentwise operations) and let $A_{\Gamma ,F}^{(\mathrm{alg})}$ be the linear operator $F^{V(\Gamma )}\to F^{V(\Gamma )}$ defined by $(A_{\Gamma ,F}^{(\mathrm{alg})}(f))(v)={\sum }_{u\in \Gamma (v)}f(u)$ for all $f\in F^{V(\Gamma )}$, $v\in V(\Gamma )$. In the case of finite graph $\Gamma$ the mapping $A_{\Gamma ,F}^{(\mathrm{alg})}$ is the well known operator defined by the adjacency matrix of $\Gamma$ (over $F$), and the theory of eigenvalues and eigenfunctions of such operator is a well-developed (at least in the case $F=\mathbb{C}$) part of the theory of finite graphs. In this paper we develope a theory of eigenvalues and eigenfunctions of $A_{\Gamma ,F}^{(\mathrm{alg})}$ for arbitrary infinite locally finite graphs $\Gamma$ (although a few results may be of interest for finite graphs) and fields $F$ with a special emphasis on the case when $\Gamma$ is connected with uniformly bounded vertex degrees and $F=\mathbb{C}$. By the author opinion, previous attempts in this direction were not quite satisfactory since were limited by consideration of rather special eigenfunctions and corresponding eigenvalues.

52 pages, in Russian; a few minor corrections; published version