On adjacency operators of locally finite graphs

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 $\Gamma$ withvertex 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 naturalcomponentwise 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 $\Gamma$, themapping $A^{({\mathrm{alg}})}_{\Gamma,F}$ is the well-known operator defined by theadjacency matrix of the graph $\Gamma$ (over $F$), andthe theory of eigenvalues and eigenfunctions of such operatorsis a well developed part of the theory of finite graphs(at least, in the case $F = \mathbb{C}$). In the present paper, wedevelop the theory of eigenvalues and eigenfunctions of the operators$A^{({\mathrm{alg}})}_{\Gamma,F}$for infinite locally finite graphs $\Gamma$ (however, some results that followmay present certain interest for the theory of finite graphs) and arbitrary fields $F$,even though in the present paper special emphasis is placedon 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.