Abstract
Let G be a graph and A(G) the adjacency matrix of G. The polynomial \(\pi (G,x)=\mathrm {per}(xI-A(G))\) is called the permanental polynomial of G, and the permanental sum of G is the summation of the absolute values of the coefficients of \(\pi (G,x)\). In this paper, we give some upper and lower bounds for the permanental sum among spiro hexagonal chains, and the corresponding extremal graphs are determined. Furthermore, we investigate the more general result about permanental sum. We obtain a lower bound for the permanental sum of bipartite graphs and the corresponding extremal graphs are also determined.
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Acknowledgements
The authors are much grateful to two anonymous referees for valuable comments on our paper, which have considerably improved the presentation of this paper. This first author is supported by the National Natural Science Foundation of China (No. 11761056), the Natural Science Foundation of Qinghai Province (2016-ZJ-947Q), and High-level Personnel of Scientific Research Project of QHMU(2016XJG07). The third author is supported by the Sungkyun research fund, Sungkyunkwan University, 2017, and National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642.
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Communicated by Xueliang Li.
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Wu, T., Ren, S. & Das, K.C. Some Extremal Graphs with Respect to Permanental Sum. Bull. Malays. Math. Sci. Soc. 42, 2947–2961 (2019). https://doi.org/10.1007/s40840-018-0642-9
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DOI: https://doi.org/10.1007/s40840-018-0642-9