Note on clique polynomials and independent set polynomial of graphs

TL;DR

This paper examines the properties of clique polynomials in graphs, providing a counterexample to a previously proposed theorem and clarifying the conditions under which it holds.

Contribution

It presents a counterexample to an existing theorem on clique polynomial edge subgraph expansion and discusses the specific cases where the theorem is valid.

Findings

Counterexample disproves the general theorem

Clarification of conditions for theorem validity

Insights into clique polynomial properties

Abstract

In 1994, Cornelis Hoede and Xueliang Li introduced the clique polynomial of a graph. Also, a theorem for the edge subgraph expansion for clique polynomials. In this note we present a counter example for it and explain which case it could be a valid theorem.