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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
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/10.4169/college.math.j.44.1.032
Page Count: 5
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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.
Copyright the Mathematical Association of America 2013