### Properties of zeta functions of quantum graphs

**Harrisson, J ***(Baylor)*

Tuesday 27 July 2010, 10:30-11.15

Seminar Room 1, Newton Institute

#### Abstract

The Ihara-Selberg zeta function plays a fundamental role in the spectral theory of combinatorial graphs. However, in contrast the spectral zeta function has remained a relatively unstudied area of analysis of quantum graphs. We consider the Laplace operator on a metric graph with general vertex matching conditions that define a self-adjoint realization of the operator. The zeta function can be constructed using a contour integral technique. In the process it is convenient to use new forms for the secular equation that extend the well known secular equation of the Neumann star graph. The zeta function is then expressed in terms of matrices defining the matching conditions at the vertices. The analysis of the zeta function allows us to obtain new results for the spectral determinant, vacuum energy and heat kernel coefficients of quantum graph which are topics of current research in their own right. The zeta function provides a unified approach which obtains general results for such spectral properties.

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