Clique roots of K4-free chordal graphs

• Main Author: Hossein Teimoori Faal
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2019-04-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: clique polynomial, clique root, chordal graph, clique decomposition
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/501

<p>The clique polynomial <span class="math"><em>C</em>(<em>G</em>, <em>x</em>)</span> of a finite, simple and undirected graph <span class="math"><em>G</em> = (<em>V</em>, <em>E</em>)</span> is defined as the ordinary generating function of the number of complete subgraphs of <span class="math"><em>G</em></span>. A real root of <span class="math"><em>C</em>(<em>G</em>, <em>x</em>)</span> is called a clique root of the graph <span class="math"><em>G</em></span>. Hajiabolhasan and Mehrabadi showed that every simple graph <span class="math"><em>G</em></span> has at least a clique root in the interval <span class="math">[ − 1, 0)</span>. Moreover, they showed that the class of triangle-free graphs has only clique roots. In this paper, we extend their result by showing that the class of <span class="math"><em>K</em>₄</span>-free chordal graphs has also only clique roots. In particular, we show that this class has always a clique root <span class="math"> − 1</span>. We conclude our paper with some interesting open questions and conjectures.</p>