  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 some examples of infinite graphs for which the
spectral multiplicity function of the adjacency operator 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, and explicit examples of infinite connected graphs which
are uniquely determined by their spectrum.
