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.
Highlights
Polynomials with only real roots are appeared in many branches of theoretical and applied mathematical sciences
The main idea is based on the graph-theoretical interpretation of the second derivative of clique polynomials
We quickely review the basics of clique polynomials
Summary
Polynomials with only real roots are appeared in many branches of theoretical and applied mathematical sciences. They showed that the class of triangle-free graphs has only clique roots This immediately implies another algebraic proof of Mantel’s theorem [1]. In [8], the author obtained a generalization of clique-rootedness of the class of triangle-free graphs He showed that an intersting subclass of chordal graphs has only clique roots. Our main motivation for introducing a new class of graphs with only clique roots is to move forward in the direction of finding a new algebraic proof of Turán’s graph theorem It seems that this idea can be more generalized to prove Turán-type extremal results. We state and prove the main result of our paper which asserts that the class of 2-connected K5-free chordal graphs has only clique roots. We conclude the paper with several interesting open questions and conjectures
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