On adjacency operators of locally finite graphs

Abstract

A graph $\Gamma$ is called locally finite if, for each vertex $v\in \Gamma$, the set $\Gamma(v)$ of its adjacent vertices is finite. For an arbitrary locally finite graph $\Gamma$ with vertex set $V(\Gamma)$ and an arbitrary 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 $A^{(\mathrm{alg})}_{\Gamma,F}$ be the linear operator $F^{V(\Gamma)} \to F^{V(\Gamma)}$ defined by $(A^{(\mathrm{alg})}_{\Gamma,F}(f))(v) = \sum_{u \in \Gamma(v)}f(u)$ for all $f \in F^{V(\Gamma)}$, $v \in V(\Gamma)$. In the case of a finite graph $\Gamma$, the mapping $A^{({\mathrm{alg}})}_{\Gamma,F}$ is the well-known operator defined by the adjacency matrix of the graph $\Gamma$ (over $F$), and the theory of eigenvalues and eigenfunctions of such operators is a well developed part of the theory of finite graphs (at least, in the case $F = \mathbb{C}$). In the present paper, we develop the theory of eigenvalues and eigenfunctions of the operators $A^{({\mathrm{alg}})}_{\Gamma,F}$ for infinite locally finite graphs $\Gamma$ (however, some results that follow may present certain interest for the theory of finite graphs) and arbitrary fields $F$, even though in the present paper special emphasis is placed on the case of a connected graph $\Gamma$ with uniformly bounded degrees of vertices and $F = \mathbb{C}$. The previous attempts in this direction were not, in the author's opinion, quite satisfactory in the sense that they have been concerned only with eigenfunctions (and corresponding eigenvalues) of rather special type.