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Spectral moments of regular graphs in terms of subgraph counts

We give equations that relate the spectral moments of a regular graph
G to the numbers of certain subgraphs that occur within G. The
equations are derived by using generating functions to count certain
closed walks in G. The subgraphs that determine the i-th spectral
moment are precisely the connected graphs of minimum degree 2 that are
induced by at least one closed walk of length i.