Emmy Noether Lectures

2008 Lecturer: Audrey A. Terras

Fun with Zeta Functions of Graphs

I will present an introduction to zeta functions of graphs along with some history and comparisons with other zetas from number theory and geometry such as Riemann’s and Selberg’s. Three kinds of graph zetas will be defined: vertex, edge and path. The basic properties will be discussed, including the Ihara formula saying that the zeta function is the reciprocal of a polynomial. I will then explore analogs of the Riemann hypothesis, zero (pole) spacings, and connections with expander graphs and quantum chaos. The graph theory version of the prime number theorem will be discussed. The graphs will be assumed to be finite undirected and possibly irregular. References include my joint papers with Harold Stark in Advances in Math.
Brief Biography:
Audrey Terras is a fellow of the American Association for the Advancement of Science and has served on that organization’s mathematics section nominating committee. She has served on the Council of the American Mathematical Society and various AMS committees and was an editor of the Transactions of the AMS. Currently she is an associate editor of book reviews for the Bulletin of the AMS and the chair of the Western Section Program Committee. She has served on various AWM committees in the past.
She has written three books: Harmonic Analysis on Symmetric Spaces and Applications, Vols. I, II, Springer-Verlag, New York, 1985, 1988 and Fourier Analysis on Finite Groups and Applications, Cambridge University Press, Cambridge, 1999. She also co-edited, with Dennis Hejhal and Peter Sarnak, the proceedings from a 1984 conference on Selberg’s trace formula. Her research interests include number theory; harmonic analysis on symmetric spaces and finite groups along with its applications; special functions; algebraic graph theory, especially zeta functions of graphs; and Selberg’s trace formula.
Current research involves finite analogues of the symmetric spaces of her Springer-Verlag volumes. This led her to work on spectra of graphs and hypergraphs attached to finite matrix groups, also coverings of graphs and their zeta and L-functions. These functions are analogues of the Riemann and Selberg zeta functions.