The Ihara-zeta function and the spectrum of the join of two semi-regular bipartite graphs

TL;DR

This paper computes the Ihara-zeta function and spanning trees of the join of two semi-regular bipartite graphs, revealing a mutual determination between their spectrum and zeta function using matrix techniques.

Contribution

It introduces a method to compute the Ihara-zeta function and spanning trees for graph joins, and shows the spectrum and zeta function uniquely determine each other.

Findings

Explicit formulas for Ihara-zeta function of graph joins

Relationship between spectrum and zeta function of graph joins

Mutual determination of spectrum and zeta function

Abstract

In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the zeta function of the join of two semi-regular bipartite graphs can determine each other.