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The eigenvalues of the adjacency operator of a finite regular graph yield a great deal of information about the graph, including bounds on its chromatic number, diameter, girth, isoperimetric constant, etc. The second largest eigenvalue is frequently of particular interest. The sum of the kth powers of these eigenvalues equals the number of closed walks of length k in the graph, so counting the latter affords us a combinatorial technique for obtaining information about the former. In this expository paper, we first illustrate this technique with the toy example of cycle graphs. We then briefly discuss how this method has been used to prove two theorems which provide lower bounds on the second largest eigenvalue of a graph, namely the Alon-Boppana theorem (for arbitrary regular graphs) as well as a theorem of Cioabă (for Cayley graphs of abelian groups).

A combinatorial trace method: Counting closed walks to assay graph eigenvalues