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Sharp Bounds on the Permanental Sum of a Graph

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Abstract

Let G be a simple undirected graph, I the identity matrix, and A(G) an adjacency matrix of G. Then the permanental sum of G equals to the permanent of the matrix \(I+A(G)\). Since the computation of the permanental sum of a graph is #P-complete, it is desirable to have good bounds. In this paper, we affirm a sharp upper bound for general graphs conjectured by Wu and So. Moreover, we prove a sharp lower bound for connected tricyclic graphs. Lastly, several unsolved problems about permanental sum are presented.

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Acknowledgements

This research of the corresponding author (TW) is supported by the National Natural Science Foundation of China (No. 11761056), the Natural Science Foundation of Qinghai Province (No. 2020-ZJ-920), and the Scientific Research Innovation Team in Qinghai Nationalities University.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, San Jose State University, San Jose, CA, 95192, USA

    Wasin So

  2. School of Mathematics and Statistics, Qinghai Nationalities University, Xining, 810007, Qinghai, People’s Republic of China

    Tingzeng Wu

  3. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, Sichuan, People’s Republic of China

    Huazhong Lü

Authors
  1. Wasin So
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  2. Tingzeng Wu
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  3. Huazhong Lü
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Correspondence to Tingzeng Wu.

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So, W., Wu, T. & Lü, H. Sharp Bounds on the Permanental Sum of a Graph. Graphs and Combinatorics 37, 2423–2437 (2021). https://doi.org/10.1007/s00373-021-02365-y

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