2022/06/04 by Faal, Hossein Teimoori
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2206.02044
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 K4-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.