On Clique Roots of Flat Graphs

TL;DR

This paper extends the understanding of clique roots in graphs, showing that certain classes of flat, K4-free graphs have clique roots in [-1,0), including the root -1, and discusses open questions in the area.

Contribution

It generalizes previous results by proving that K4-free flat graphs also have clique roots in [-1,0), expanding the classes of graphs known to have this property.

Findings

K4-free flat graphs have clique roots in [-1,0)

The class of K4-free flat graphs without isolated edges has -1 as a clique root

Open questions and conjectures about clique roots are presented

Abstract

A complete subgraph of a given graph is called a clique. A clique Polynomial of a graph is a generating function of the number of cliques in $G$. A real root of the clique polynomial of a graph $G$ is called a \emph{clique root} of $G$. \\ Hajiabolhassan and Mehrabadi showed that the clique polynomial of any simple graph has a clique root in $[-1,0)$. As a generalization of their result, the author of this paper showed that the class of $K_{4}$-free connected chordal graphs has also only clique roots. \\ A given graph $G$ is called flat if each edge of $G$ belongs to at most two triangles of $G$. In answering the author's open question about the class of \emph{non-chordal} graphs with the same property of having only c;ique roots, we extend the aforementioned result to the class of $K_{4}$-free flat graphs. In particular, we prove that the class of $K_{4}$-free flat graphs without isolated edges has $r=-1$ as one of its clique roots. We finally present some interesting open questions and conjectures regarding clique roots of graphs.