Clique Polynomials and Chordal Graphs

TL;DR

This paper studies the roots of clique polynomials in chordal graphs, proving that certain classes of these graphs have only real clique roots, including the root -1, and discusses related open questions.

Contribution

It generalizes previous results by showing that K4-free chordal graphs have only clique roots, including the root -1, expanding understanding of clique polynomial roots in graph theory.

Findings

K4-free chordal graphs have only clique roots

Such graphs always have -1 as a clique root

The paper proposes open questions and conjectures

Abstract

The ordinary generating function of the number of complete subgraphs of $G$ is called a clique polynomial of $G$ and is denoted by $C(G,x)$. A real root of $C(G,x)$ is called a clique root of the graph $G$. Hajiabolhasan and Mehrabadi showed that the clique polynomial has always a real root in the interval $[-1,0)$. Moreover, they showed that the class of triangle-free graphs has only clique roots. Here, we generalize their result by showing that the class of $K_4$-free chordal graphs has also only clique roots. Moreover, we show that this class has always a clique root $-1$. We finally conclude the paper with several important questions and conjectures.