Vol. 1, No. 2, 2008

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The coefficients of the Ihara zeta function

Geoffrey Scott and Christopher Storm

Vol. 1 (2008), No. 2, 217–233

In her Ph.D. Thesis, Czarneski began a preliminary study of the coefficients of the reciprocal of the Ihara zeta function of a finite graph. We give a survey of the results in this area and then give a complete characterization of the coefficients. As an application, we give a (very poor) bound on the number of Eulerian circuits in a graph. We also use these ideas to compute the zeta function of graphs which are cycles with a single chord. We conclude by posing several questions for future work.

Ihara zeta, polynomial coefficient, graph zeta, Eulerian circuit, graph, digraph, oriented line graph
Mathematical Subject Classification 2000
Primary: 00A05
Received: 29 October 2007
Revised: 28 February 2008
Accepted: 6 May 2008
Published: 1 July 2008

Communicated by Andrew Granville
Geoffrey Scott
Department of Mathematics
2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States
Christopher Storm
Department of Mathematics and Computer Science
111 Alumnae Hall
Adelphi University
Garden City, NY 11530
United States