Complete equitable decompositions

A classical result in spectral graph theory states that if a graph G has an equitable partition π then the eigenvalues of the divisor graph $G_{\pi }$ are a subset of its eigenvalues, i.e. $\sigma \left(G_{\pi }\right)\subseteq \sigma \left(G\right)$. A natural question is whether it is possible to recover the remaining eigenvalues $\sigma \left(G\right)-\sigma \left(G_{\pi }\right)$ in a similar manner. Here we show that any weighted undirected graph with nontrivial equitable partition can be decomposed into a number of subgraphs whose collective spectra contain these remaining eigenvalues. Using this decomposition, which we refer to as a complete equitable decomposition, we introduce an algorithm for finding the eigenvalues of an undirected graph (symmetric matrix) with a nontrivial equitable partition. Under mild assumptions on this equitable partition we show that we can find eigenvalues of such a graph faster using this method when compared to standard methods. This is potentially useful as many real-world data sets are quite large and have a nontrivial equitable partition.