  It is common knowledge that a key dynamical characteristic of a network is
its spectrum (the collection of all eigenvalues of the network's weighted
adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce
a network, considered as a graph, to a smaller network with fewer vertices and
edges while preserving the spectrum (or spectral information) of the original
network. This procedure allows for the introduction of new equivalence
relations between networks, where two networks are spectrally equivalent if
they can be reduced to the same network. Additionally, using this theory it is
possible to establish whether a network, modeled as a dynamical system, has a
globally attracting fixed point (is strongly synchronizing). In this paper we
further develop this theory of isospectral network transformations and
demonstrate that our procedures are applicable to families of parameterized
networks and networks of arbitrary size.
