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Unimodality of independence polynomials of rooted products of graphs

Published online by Cambridge University Press:  04 June 2019

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)

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Alavi, Y., Malde, P. J., Schwenk, A. J. and Erdős, P.. The vertex independence sequence of a graph is not constrained. Congr. Numer. 58 (1987), 1523.Google Scholar
2Bahls, P.. On the independence polynomils of path-like graphs. Australas. J. Combin. 53 (2012), 318.Google Scholar
3Bahls, P. and Salazar, N.. Symmetry and unimodality of independence polynomials of path-like graphs. Australas. J. Combin. 47 (2010), 165176.Google Scholar
4Bahls, P., Ethridge, B. and Szabo, L.. Unimodality of the independence polynomials of non-regular caterpillars. Australas. J. Combin. 71 (2018), 104112.Google Scholar
5Bencs, F.. On trees with real rooted independence polynomial. Discrete Math. 341 (2018), 33213330.CrossRefGoogle Scholar
6Brenti, F.. Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update. Contemp. Math. 178 (1994), 417441.Google Scholar
7Chudnovsky, M. and Seymour, P.. The roots of the independence polynomial of a clawfree graph. J. Combin. Theory Ser. B 97 (2007), 350357.Google Scholar
8Galvin, D. and Hilyard, J.. The independent set sequence of some families of trees, arxiv: 1701.02204, (2017).Google Scholar
9Godsil, C. D. and McKay, B. D.. A new graph product and its spectrum. Bull. Austral. Math. Soc. 18 (1978), 2128.CrossRefGoogle Scholar
10Gutman, I. and Harary, F.. Generalizations of the matching polynomial. Utilitas Math. 24 (1983), 97106.Google Scholar
11Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge: Cambridge University Press, 1952).Google Scholar
12Heilmann, O. J. and Lieb, E. H.. Theory of monomer-dimer systems. Comm. Math. Phys. 25 (1972), 190232.Google Scholar
13Huh, J.. Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs. J. Amer. Math. Soc. 25 (2012), 907927.CrossRefGoogle Scholar
14Levit, V. E. and Mandrescu, E.. On well-covered trees with unimodal independence polynomials. Congr. Numer. 159 (2002), 193202.Google Scholar
15Levit, V. E. and Mandrescu, E.. Very well-covered graphs with log-concave independence polynomials. Carpathian J. Math. 20 (2004), 7380.Google Scholar
16Levit, V. E. and Mandrescu, E.. The independence polynomial of a graph – a survey, Proceedings of the 1st International Conference on Algebraic Informatics,pp. 233254, Aristotle Univ. Thessaloniki, Thessaloniki, (2005).Google Scholar
17Levit, V. E. and Mandrescu, E.. Independence polynomials of well-covered graphs: generic counterexamples for the unimodality conjecture. European J. Combin. 27 (2006), 931939.Google Scholar
18Levit, V. E. and Mandrescu, E.. On the roots of independence polynomials of almost all very well-covered graphs. Discrete Appl. Math. 156 (2008), 478491.CrossRefGoogle Scholar
19Mandrescu, E.. Building graphs whose independence polynomials have only real roots. Graphs Combin. 25 (2009), 545556.CrossRefGoogle Scholar
20Mandrescu, E. and Spivak, A.. Maximal trees with log-concave independence polynomials. Notes on Number Theory and Discrete. Math. 22 (2016), 4453.Google Scholar
21Read, R. C.. An introduction to chromatic polynomials. J. Combin. Theory 4 (1968), 5271.CrossRefGoogle Scholar
22Stanley, R. P.. Log-concave and unimodal sequences in algebra, combinatorics and geometry. Ann. New York Acad. Sci. 576 (1989), 500534.CrossRefGoogle Scholar
23Wagner, D. G.. Zeros of reliability polynomials and f-vectors of matroids. Combin. Probab. Comput. 9 (2000), 167190.CrossRefGoogle Scholar
24Wang, Y. and Zhu, B.-X.. On the unimodality of independence polynomials of some graphs. European J. Combin. 32 (2011), 1020.CrossRefGoogle Scholar
25Welsh, D.. Matriod theory (London/New York: Academic Press, 1976).Google Scholar
26Zhu, B.-X.. Clique cover products and unimodality of independence polynomials. Discrete Appl. Math. 206 (2016), 172180.CrossRefGoogle Scholar
27Zhu, B.-X.. Unimodality of independence polynomials of the incidence product of graphs. Discrete Math. 341 (2018), 23592365.Google Scholar
28Zhu, Z.-F.. The unimodality of independence polynomials of some graphs. Australas. J. Combin. 38 (2007), 2733.Google Scholar
29Zhu, B.-X. and Chen, Y.. Log-concavity of independence polynomials of some kinds of trees. Appl. Math. Comput. 342 (2019), 3544.CrossRefGoogle Scholar
30Zhu, B.-X. and Lu, Q.. Unimodality of the independence polynomials of some composite graphs. Filomat. 31 (2017), 629637.CrossRefGoogle Scholar