A combinatorial trace method: Counting closed walks to assay graph eigenvalues
Keith C. D'Souza, Mike Krebs · Rocky Mountain Journal of Mathematics · 2013
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 Cioabȃ (for Cayley graphs of abelian groups). Introduction.While the eigenvalues of a finite graph do not completely determine the graph, as shown by the existence of nonisomorphic isospectral graphs, they do convey a tremendous amount of information about the graph.Virtually every graph invariant is in some way intimately related to the graph's spectrum, i.e., its multiset of eigenvalues, counted with multiplicity.Examples include the chromatic number, girth, diameter, isoperimetric constant, etc., see [3, 10, 11] for comprehensive surveys.For a regular graph, the second largest eigenvalue λ 1 is frequently of primary interest.The following theorem of Chung [4] is typical: If X is a d-regular graph with n vertices, then diam (X) ≤ log(n -1)/ log(d/λ) , where λ is the eigenvalue of X with second-largest absolute value.See [5] for a more comprehensive survey of results along these lines.Computing, or even finding, good bounds for the eigenvalues of a family of graphs can be quite difficult.The purpose of this expository