Infinite locally finite connected graphs with countable complements in C of the sets of eigenvalues

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 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 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 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 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 C of at least one infinite locally finite connected graph.