  We define a graph to be $S$-regular if it contains an equitable partition
given by a matrix $S$. These graphs are generalizations of both regular and
bipartite, biregular graphs. An $S$-regular matrix is defined then as a matrix
on an $S$-regular graph consistent with the graph's equitable partition. In
this paper we derive the limiting spectral density for large, random
$S$-regular matrices as well as limiting functions of certain statistics for
their eigenvector coordinates as a function of eigenvalue. These limiting
functions are defined in terms of spectral measures on $S$-regular trees. In
general, these spectral measures do not have a closed-form expression; however,
we provide a defining system of polynomials for them. Finally, we explore
eigenvalue bounds of $S$-regular graph, proving an expander mixing lemma,
Alon-Bopana bound, and other eigenvalue inequalities in terms of the
eigenvalues of the matrix $S$.
