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Unimodality of independence polynomials of rooted products of graphs

Part of: Combinatorics Graph theory

Published online by Cambridge University Press:  04 June 2019

Bao-Xuan Zhu  and
Qingxiu Wang
Bao-Xuan Zhu
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou221116, P. R. China (bxzhu@jsnu.edu.cn; baoxuan.zhu@ucl.ac.uk)
Qingxiu Wang
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou221116, P. R. China (bxzhu@jsnu.edu.cn; baoxuan.zhu@ucl.ac.uk)
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Abstract

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

Keywords

Unimodalityreal zerosindependence polynomialsclaw-free graphsrooted product

MSC classification

Primary: 05C69: Dominating sets, independent sets, cliques
Secondary: 05A20: Combinatorial inequalities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2573 - 2585
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Copyright
Copyright © Royal Society of Edinburgh 2019

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