Co-adjoint polynomial

TL;DR

This paper introduces a new graph polynomial derived from a specific recursion, linking it to the Tutte polynomial and comparing it with other well-known graph polynomials like chromatic and matching polynomials.

Contribution

It identifies a novel graph polynomial within a family of four related polynomials, establishing its properties and connection to the Tutte polynomial.

Findings

The polynomial is of exponential type.

It is a specialization of the Tutte polynomial.

Shares properties with chromatic, matching, and adjoint polynomials.

Abstract

In this note we study a certain graph polynomial arising from a special recursion. This recursion is a member of a family of four recursions where the other three recursions belong to the chromatic polynomial, the modified matching polynomial, and the adjoint polynomial, respectively. The four polynomials share many common properties, for instance all of them are of exponential type, i. e., they satisfy the identity $$\sum_{S\subseteq V(G)}f(G[S],x)f(G[V\setminus S],y)=f(G,x+y)$$ for every graph $G$. It turns out that the new graph polynomial is a specialization of the Tutte polynomial.