Abstract:
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.
