A generalization of zeta function of a graph and its application

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540176
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Oyama National College of Technology
• Principal Investigator: SATO Iwao 小山工業高等専門学校, 一般科, 教授 (70154036)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: グラフ / ゼータ関数 / 量子ウォーク

Outline of Final Research Achievements

We presented determinant expressions for the weighted Bartholdi zeta function of a digraph and the matrix-weighted Ihara $L$-function of a graph. Moreover, we presented a determinant expression of an edge $L$-function of a graph. Furthermore, we gave a determinant expression for the generalized Bartholdi zeta function of a bipartite graph, and then obtained a determinant expression for the generalized Bartholdi zeta function of a hypergraph. We considered the edge zeta function and the edge $L$-function of a hypergraph.
We presented the characteristic polynomials of the Grover matrix and its positive support, the positive support of its square by using determinant expressions of zeta functions of a graph. As applications, we determined their spectra. Furthermore, we define a zeta function of a graph with the positive support of the square of the Grover matrix as an edge matrix, and treated the Euler product, the determinant expression, the pole and the radius of convergence of it.

Free Research Field

グラフ理論