Abstract:
In a previous paper, the author proved that non-eigenvalues of the adjacency operator of an infinite locally finite connected graph over a field of characteristic 0 can be only algebraic over the prime subfield of the field elements (in particular, only algebraic numbers when the field is $\mathbb{C}$). 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 infiniely many algebraic numbers are not eigenvalues of its adjacency operator over $\mathbb{C}$. More exactly, 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 the paper a necessary condition (based on results of the mentioned 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.
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