  We investigate connections between the symmetries (automorphisms) of a graph
and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial
automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix
$M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The
result of this decomposition is a number of strictly smaller matrices whose
collective eigenvalues are the same as the eigenvalues of the original matrix
$M$. Some of the matrices that can be decomposed are the graph's adjaceny
matrix, Laplacian matrix, etc. Because this decomposition has connections to
the theory of equitable partitions it is referred to as an equitable
decomposition. Since the graph structure of many real-world networks is quite
large and has a high degree of symmetry, we discuss how equitable
decompositions can be used to effectively bound both the network's spectral
radius and spectral gap, which are associated with dynamic processes on the
network. Moreover, we show that the techniques used to equitably decompose a
graph can be used to bound the number of simple eigenvalues of undirected
graphs, where we obtain sharp results of Petersdorf-Sachs type.
