A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta Function for Graphs

Binary linear codes via 4D discrete Ihara-Selberg function

We express the weight enumerator of each binary linear code, in particular the Ising partition function of an arbitrary finite graph, as a formal infinite product. An analogous result was obtained by Feynman and Sherman in the beginning of the 1960's for the special case of the Ising partition function of planar graphs. A product expression is an important step towards understanding the logarithm of the Ising partition function, for general graphs and in particular for cubic 3D lattices.

Ihara Zeta functions of infinite weighted graphs

The theory of Ihara zeta functions is extended to infinite graphs which are weighted and of finite total weight. In this case one gets meromorphic instead of rational functions and the classical determinant formulas of Bass and Ihara hold true with Fredholm determinants.

超越基线崩溃：Ihara理论中的新不规则边空间不变量

Beyond Bass Collapse: New Irregular Edge-Space Invariants in Ihara Theory

本文通过边反转诱导的对称/反对称分裂，揭示了Ihara理论中普通和反对称边图谱的两个标准对角部分，并通过Schur补论证得到修正行列式，展示了不规则边空间不变量。

Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus g as a linear combination of 2^{2g} Pfaffians of Dirac operators. Let G=(V,E) be a finite graph embedded in a closed Riemann surface X of genus g, x_e the collection of independent variables associated with each edge e of G (collected in one vector variable x) and S the set of all 2^{2g} Spin-structures on X. We introduce 2^{2g} rotations rot_s and (2|E| times 2|E|) matrices D(s)(x), s in S, of the transiti…

The Zeta Function of a Hypergraph

We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Solé. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.

A trace on fractal graphs and the Ihara zeta function

Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, …