One more remark on the adjoint polynomial

TL;DR

This paper reveals a simple relationship between the adjoint polynomial of a graph and the independence polynomial of another graph, allowing the application of existing theories to study adjoint polynomials.

Contribution

It establishes a transformation linking adjoint and independence polynomials, enabling new proofs of existing theorems and expanding analytical tools.

Findings

Adjoint polynomial is a transformation of the independence polynomial of another graph.

New proofs of theorems by Liu and Csikvári are provided.

The relationship broadens the analytical framework for studying graph polynomials.

Abstract

The adjoint polynomial of $G$ is \[h(G,x)=\sum_{k=1}^n(-1)^{n-k}a_k(G)x^k,\] where $a_k(G)$ denotes the number of ways one can cover all vertices of the graph $G$ by exactly $k$ disjoint cliques of $G$. In this paper we show the the adjoint polynomial of a graph $G$ is a simple transformation of the independence polynomial of another graph $\widehat{G}$. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we a give new proofs of several theorems of R. Liu and P. Csikv\'{a}ri.