December 1-5, 2008, Moscow, Russia
Oyama National College of Technology, Japan
isato (at) oyama-ct.ac.jp
The Ihara zeta function of a graph and its generalization
At first, we state some basic definitions and notation used in this talk, and give a survey on Ihara zeta functions of graphs. Ihara zeta functions of regular graphs were started from number theory, and they have fuitful inter- esting properties related to many fields in Mathematics: The rationality, the functional equation, the analogue of the Riemann hypothesis, the analogue of the Prime Number Theorem. Furthermore, we state on Ihara zeta functions of irregular graphs and their weighted versions.
Secondly, we talk about another type zeta function of a graph, i.e., the Bartholdi zeta function. This is a two-variable zeta function of a graph, and was defined by Bartholdi.
Thirdly, we state a digraph version and a weighted version of the Bartholdi zeta function of a graph.
Finally, we define a general Bartholdi zeta function of a digraph which contains five zeta functions stated in the above. Thus, we present results on five zeta functions as corollaries of a result of a general Bartholdi zeta function of a digraph.
If we have any time to spare, then we will state a generalization of Bartholdi zeta function of a graph to an infinite graph.
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