Graphs and Generalized Witt identities

TL;DR

This paper explores determinantal identities related to Ihara and Bowen-Lanford zeta functions of graphs, revealing their connections to Lie superalgebras, combinatorics, and coloring problems, and generalizing classical identities.

Contribution

It demonstrates that Ih and BL identities are special cases of Witt identities, generalizes classical identities, and provides new interpretations linked to Lie superalgebras and graph colorings.

Findings

Ih and BL identities satisfy generalizations of Strehl and Carlitz-Metropolis-Rota relations.

They can be interpreted as denominator identities of free Lie superalgebras.

New interpretations of the Ihara and Bowen-Lanford zeta functions are provided.

Abstract

This paper is about the determinantal identities associated with the Ihara (Ih) zeta function of a non directed graph and the Bowen-Lanford (BL) zeta function of a directed graph. They will be called the Ih and the BL identities in this paper. We show that the Witt identity (WI) is a special case of the BL identity and inspired by the links the WI has with Lie algebras and combinatorics we investigate similar aspects of the Ih and BL identities. We show that they satisfy generalizations of the Strehl identity and Carlitz, Metropolis-Rota relations and each one of them can be interpreted as the denominator (or generalized Witt) identity of a free Lie superalgebra. Also, they can be associated to a coloring problem. New interpretations of the Ih and BL zeta functions are presented.