Clique roots of K4-free chordal graphs

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