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Unimodality of Independence Polynomials of the Cycle Cover Product of Graphs

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Abstract

An independent set in a graph G is a set of pairwise non-adjacent vertices. Let ik(G) denote the number of independent sets of cardinality k in G. Then its generating function

$$I(G;x) = \sum\limits_{k = 0}^{\alpha (G)} {{i_k}(G){x^k}} $$

is called the independence polynomial of G (Gutman and Harary, 1983). In this paper, we introduce a new graph operation called the cycle cover product and formulate its independence polynomial. We also give a criterion for formulating the independence polynomial of a graph. Based on the exact formulas, we prove some results on symmetry, unimodality, reality of zeros of independence polynomials of some graphs. As applications, we give some new constructions for graphs having symmetric and unimodal independence polynomials.

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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, P. R. China

    Bao Xuan Zhu

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  1. Bao Xuan Zhu
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Correspondence to Bao Xuan Zhu.

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Supported by National Natural Science Foundation of China (Grant Nos. 11971206, 12022105) and Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province (Grant No. BK20200048)

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Zhu, B.X. Unimodality of Independence Polynomials of the Cycle Cover Product of Graphs. Acta. Math. Sin.-English Ser. 38, 858–868 (2022). https://doi.org/10.1007/s10114-022-0503-1

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  • DOI: https://doi.org/10.1007/s10114-022-0503-1

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