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.Comment: 24 pages, 3 figure