Spectral multiplicity functions of adjacency operators of graphs and cospectral infinite graphs

The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this article is to review a small number of examples of infinite graphs $G = (V,E)$ for which the spectral multiplicity function of the adjacency operator $A_G$ of $G$ has been determined. The second purpose of this article is to show explicit examples of infinite connected graphs which are cospectral, i.e., which have unitarily equivalent adjacency operators