# Clique polynomials of $2$-connected $K_{5}$-free chordal graphs

### Abstract

In this paper, we give a generalization of the author's previous result about real rootedness of clique polynomials of connected $K_{4}$-free chordal graphs to the class of $2$-connected $K_{5}$-free chordal graphs. The main idea is based on the graph-theoretical interpretation of the second derivative of clique polynomials. Finally, we conclude the paper with several interesting open questions and conjectures.

### References

J. A. Bondy, U. S. R. Murty, Graph theory, Springer GTM 244 (2008).

P. Branden, Unimodality, log-concavity, real-rootedness and beyond, Handbook of Enumerative Combinatorics, CRC Perss (2018).

L. Comet, Advanced combinatorics, 200. Reidel, Dordrecht-Boston (1974).

H. Hajiabolhassan, M. L. Mehrabadi, On clique polynomials, Australasian Journal of Combinatorics 18 (1998) 313–316.

P. Haxell, A. Kostochka, S. Thomasse, Packing and covering triangles in K4-free planar graphs, Discrete Applied Mathematics 28 (2012) 653–662.

X. Li, I. Gutman, A unified approach to the first derivatives of graph polynomials, Discrete Applied Mathematics 587 (1995) 293–297.

T. A. McKee, F. R. McMorris, Topics in intersection graph theory (Monographs on Discrete Mathematics and Applications), Society for Industrial and Applied Mathematics (1987).

H. Teimoori, Clique roots of K4-free chordal graphs, Electronic Journal of Graph Theory and Applications 7(1) (2010) 105–111.

A. A. Zykov, On some properties of linear complexes, Mat. Sbornik N.S. 24(66) (1949) 163–188.