Eigenvalues and eigenfunctions of locally finite graphs

A graph is called locally finite if degrees of all its vertices are finite. For a locally finite graph $\Gamma$ and a field $F$, eigenvalues and their corresponding eigenfunctions of $\Gamma$ over $F$ are defined as eigenvalues and their corresponding eigenfunctions of adjacency matrix of the graph $\Gamma$ over the field $F$, acting in the natural way on the space of all $F$-valued functions on the vertex set of $\Gamma$. Eigenvalues and eigenfunctions of finite graphs (at least, over the field $\mathbf{C}$) are subjects of consideration in a well-developed part of the theory of finite graphs. But for a number of areas of mathematics, just eigenvalues and eigenfunctions of infinite locally finite connected graphs are of interest. In the talk, a theory of eigenvalues and eigenfunctions of such graphs over fields is given. A special emphasis will be placed on the case of fields of characteristic $0$ and especially on the cases of fields $\mathbf{C}$ and ${\mathbf Q}(x)$. One of the consequences of the theory: if char$(F) = 0$, then any element of $F$ which is transcendental over the prime subfield of $F$ is an eigenvalue (over $F$) of each infinite locally finite connected graph.