  We describe the complete spectra of Laplacian, signless Laplacian, and
adjacency matrices associated with the commuting graphs of a finite group using
group theoretic information. We provide a method to find the center of a group
by only using the Laplacian of the commuting graph of the group. The graph
invariants (such as diameter, clique number, and mean distance) of the
commuting graph associated with a finite group are determined. We produce a
number of examples to illustrate the applications of our results.
