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The Combinatorial Trace Method in Action

Mike Krebs and Natalie Martinez
The College Mathematics Journal
Vol. 44, No. 1 (January 2013), pp. 32-36
DOI: 10.4169/college.math.j.44.1.032
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Page Count: 5
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The Combinatorial Trace Method in Action
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Summary On any finite graph, the number of closed walks of length k is equal to the sum of the kth powers of the eigenvalues of any adjacency matrix. This simple observation is the basis for the combinatorial trace method, wherein we attempt to count (or bound) the number of closed walks of a given length so as to obtain information about the graph’s eigenvalues, and vice versa. We give a brief overview and present some simple but interesting examples. The method is also the source of interesting, accessible undergraduate projects.

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