Related papers: The Ihara expression of a generalization of the we…
We consider the generalized weighted zeta function for a finite digraph, and show that it has the Ihara expression, a determinant expression of graph zeta functions, with a certain specified definition for inverse arcs. A finite digraph in…
The Ihara expression of a weighted zeta function for a general finite digraph is given. It unifies all the Ihara expressions obtained for known zeta functions for finite digraphs. Any digraph in this paper permits multi-edges and…
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…
The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment…
We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product which is regarded as the Euler…
We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and…
From the viewpoint of quantum walks, the Ihara zeta function of a finite graph can be said to be closely related to its evolution matrix. In this note we introduce another kind of zeta function of a graph, which is closely related to, as to…
The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.
The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of…
We define a zeta function of a graph by using the time evolution matrix of a general coined quantum walk on it, and give a determinant expression for the zeta function of a finite graph. Furthermore, we present a determinant expression for…
In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this…
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
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…
We define a new weighted zeta function for a finite graph and obtain its determinant expression. This result gives the characteristic polynomial of the transition matrix of the Szegedy walk on a graph.
Chinta, Jorgenson and Karlsson introduced a generalized version of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. On the other hand, Konno and Sato obtained a formula of the…
We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function…
We establish a generalization of the second weighted zeta function of a graph to the case of quaternions. For an arc-weighted graph whose weights are quaternions, we define the second weighted zeta function by using the Study determinant…
We define a zeta function of a finite graph derived from time evolution matrix of quantum walk, and give its determinant expression. Furthermore, we generalize the above result to a periodic graph.
The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\textit{J. Combin. Theory Ser. B}, 74 (1998), pp.…
In this paper, we present formulas for the edge zeta function and the second weighted zeta function with respect to the group matrix of a finite abelian group $\Gamma $. Furthermore, we give another proof of Dedekind Theorem for the group…