Identities of the multi-variate independence polynomials from heaps theory

TL;DR

This paper explores multi-variate independence polynomials using heaps theory, deriving identities through combinatorial bijections and revealing new identities involving bipartite subgraphs.

Contribution

It introduces novel multi-variate identities for independence polynomials derived from heaps theory and combinatorial bijections.

Findings

Derived multi-variate Godsil type identity

Established fundamental identity for independence polynomials

Presented a new identity involving connected bipartite subgraphs

Abstract

We study and derive identities for the multi-variate independence polynomials from the perspective of heaps theory. Using the inversion formula and the combinatorics of partially commutative algebras we show how the multi-variate version of Godsil type identity as well as the fundamental identity can be obtained from weight preserving bijections. Finally, we obtain a new multi-variate identity involving connected bipartite subgraphs similar to the Christoffel-Darboux type identities obtained by Bencs.