Abstract
In the author’s previous paper on adjacency operators of locally finite graphs, it was proven that, for any infinite locally finite graph and any field of characteristic zero, only algebraic over the prime subfield of the field elements (in particular, only algebraic numbers when the field is \(\mathbb{C}\) ) may not be eigenvalues of the adjacency operator of the graph over the field. There were also given examples of infinite locally finite connected graphs for which certain algebraic numbers are not eigenvalues of their adjacency operators over \(\mathbb{C}\) . In the present paper, we give examples of infinite locally finite connected graphs for each of which infinitely many algebraic numbers are not eigenvalues of its adjacency operator over \(\mathbb{C}\) . More precisely, for every prime integer \(p\) , we construct an infinite locally finite connected graph such that no positive integer multiple of \(p\) is an eigenvalue of the adjacency operator over \(\mathbb{C}\) of the graph. In addition, in this paper, a necessary condition (based on the results of the abovementioned previous paper) is given for an algebraic number not to be an eigenvalue of the adjacency operator over \(\mathbb{C}\) of at least one infinite locally finite connected graph.
