Zeta-equivalent digraphs: Simultaneous cospectrality

TL;DR

This paper introduces a zeta function for digraphs that captures spectra of various matrix combinations, unifying several spectral equivalences and providing methods for constructing zeta-equivalent digraphs.

Contribution

It defines a new zeta function for digraphs that characterizes spectral properties and introduces a framework for understanding and constructing zeta-equivalent digraphs.

Findings

Zeta function determines spectra of all linear combinations of key matrices.

Zeta-equivalence encompasses multiple spectral equivalences like cospectrality.

Provides a method to construct zeta-equivalent digraphs.

Abstract

We introduce a zeta function of digraphs that determines, and is determined by, the spectra of all linear combinations of the adjacency matrix, its transpose, the out-degree matrix, and the in-degree matrix. In particular, zeta-equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency, the Laplacian, the signless Laplacian, and the normalized Laplacian matrix, respectively. In addition, we express zeta-equivalence in terms of Markov chains and in terms of invasions where each edge is replaced by a fixed digraph. We finish with a method for constructing zeta-equivalent digraphs.