The aim of the session is to discuss current work on the Ihara-Selberg zeta functions attached to graphs and related topics such as Ramanujan graphs, the trace formula on trees. The hope is to emphasize connections between various fields such as graph theory, topology, mathematical physics, number theory, dynamical systems. One example is the connection between graph zeta functions and Jones polynomials of knots found by Lin and Wang.

The following is a list of those invitees who have indicated that they will attend the meeting. Please send in your title and abstract this week if you haven't already done so. The deadline (March 15) approaches. Please note that March 15 is also the deadline for registration.

Title: Ramanujan type graphs and bigraphs

Abstract: We will show that quotients of the Bruhat-Tits building of $SL_2(\mathbb{Q}_p)$ form an infinite family of graphs which are almost Ramanujan. We will also investigate the Bruhat-Tits tree associated with $U_3(\mathbb{Q}_p)$ and show why one should be able to estimate its spectrum.

Title: Generalized Green and Zeta functions of graphs

Abstract: Let $X$ be a $d$-regular graph, with fixed vertices $x,y$. A path $\pi$ from $x$ to $y$ in $X$ has a \emph{length} $|\pi|$ and a \emph{bump count} $bc(\pi)$ counting the number of mutually inverse consecutive edges $\pi$ contains.

The standard Green function of $X$ is the power series $G(t)=\sum_\pi t^{|\pi|}$, the sum ranging over paths from $x$ to $y$. A generalized power series is $F(t,u)=\sum_\pi t^{|\pi|}u^{bc(\pi)}$. I will give a functional equation relating $G(t)$ and $F(t,u)$, and show its various applications; in particular, it gives the celebrated "Grigorchuk formula".

Assume now that $X$ is finite. A cycle in $X$ is an equivalence class of closed paths in $X$, under change of starting point. The Zeta function of $X$ is $\sum_{[\pi]}t^{|\pi|}, the sum ranging over all cycles all of whose representatives have no bump. This can be generalized to $\sum_{[\pi]}t^{\pi}u^{cbc(\pi)}$, where $cbc(\pi)$ is the number of bumps on $\pi$ viewed as a cyclic sequence of edges.

Again there is a functional equation connecting these two series; it implies the evaluation formulas of the classical Zeta function by Ihara, Bass, Foata and Zeilberger.

Title: Zeta functions of infinite graphs

Abstract: Start with a finite graph $X$. Take an infinite regular covering $Y$ of $X$, with covering group $\Gamma$. Using the trace on the von Neumann algebra associated to $\Gamma$, there is an $L^2$ zeta function for $Y$ which enjoys many of the properties of the Ihara-Hashimoto-Bass zeta function for finite graphs, including a version of the rationality formula. For families of finite graphs covering $X$, the normalized zeta functions of the finite graphs converge to the $L^2$-zeta function for $Y$.

Title: Title: Zeta functions of graphs, Shimura varieties, and dynamical systems

Abstract: This talk surveys the relations, known or conjectural, connecting zeta functions defined for three classes of objects - (hyper)graphs, modular varieties, and dynamical systems.

Title: Ramanujan hypergraphs

Abstract: A hypergraph is a higher dimensional generalization of a graph. In this talk we introduce the concept of Ramanujan hypergraphs and discuss explicit construction of such hypergraphs. These are extensions of Ramanujan graphs.

Title: Random walks on knot diagrams

Abstract: We will discuss some features of the study of a model of random walks on knot diagrams. This model relates the Alexander and Jones polynomials in knot theory with the Ihara-Selberg type zeta function in number theory and graph theory.

Title: Laplacians related to Zeta functions and an application to cogrowth of graphs.

Abstract: The concept of cogrowth of groups goes back to Ol'shanskii's settling of the von Neumann conjecture in 1984. Ol'shanskii constructed a group which was non-amenable (equivalently, its cogrowth constant was strictly less than the growth constant of its corresponding free covering) but was not an extension of a free group. The notion of amenability for finitely generated groups has been extended to arbitrary graphs and, by a suitable definition of cogrowth constant, we prove that a graph which is the cover of a finite graph is amenable if and only if its cogrowth constant equals the growth constant of its free cover. The proof uses harmonic functions with respect to operators of the form (I-uA+u^2Q) which are, by Bass' theorem, related to zeta functions on graphs.

Title: Isospectrality Conditions for Regular Graphs

Abstract: In 1994, Hubert Pesce proved that two compact hyperbolic surfaces are (strongly) isospectral if and only if their covering groups are representation equivalent as subgroups of the automorphism group of hyperbolic 2-space. The proof depends on a certain length spectrum that can be described geometrically on the surface or algebraically in the covering group.

We discuss the analogous length spectrum on regular graphs, looking at it combinatorially on the graph and algebraically in the covering group. We also note that our length spectrum appears in the Ihara zeta function.

and Hirobumi Mizuno, Meisei University, email: mizuno@ei.meisei-u.ac. jp

Title: L-functions and the Selberg trace formula for semiregular bipartite graphs

Abstract: We give a decomposition formula for the L-function of a semiregular bipartite graph G. Furthermore, we present the Selberg trace formula for the above L-function of G.

Title: "A New Kind of Zeta Function: When Number Theory Meets Graph Theory"

Abstract: The tree of zeta functions has many branches including those from number theory (Riemann and Dedekind zeta functions), spectral geometry of manifolds (Selberg's zeta function), and graph theory (Ihara's zeta function). It is also possible to mix in group representations and obtain L-functions. Applications include analogues of the prime number theorem and analogues of the work on what is now called quantum chaos - the statistics of energy levels of various non-classical physical systems. For example, the poles of the Ihara zeta function of a connected regular graph satisfy the Riemann hypothesis if and only if the graph is a Ramanujan graph (meaning that the second largest eigenvalue of the adjacency matrix, in absolute value, is in some sense smallest possible). In this talk I will compare the various sorts of zetas and investigate properties and applications of 3 types of graph zeta functions (the vertex or Ihara zeta, the edge and the path zetas) considered in Stark and Terras, Advances in Math., Vol. 121 (1996) and Vol. 154 (2000).

Title: New group theoretical constructions of Ramanujan graphs and expanders and applications

Title: Chaotic properties of quotients of trees (joint work with Audrey Terras)

Abstract: The geodesic flow on the k-regular tree produces induces a trajectory on the graph corresponding to a quotient of the tree. We look at the self correlation of the induced flow and show that it exhibits chaotic properties.