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On the permanental sum of bicyclic graphs

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Abstract

Let A(G) be the adjacency matrix of a graph G. The permanental polynomial of G is defined as \(\pi (G,x)={\mathrm {per}}(xI-A(G))\). The permanental sum of G can be defined as the sum of the absolute values of the coefficients of \(\pi (G,x)\). In this paper, we investigate the properties of the permanental sum of bicyclic graphs. We present upper and lower bounds of the permanental sum of bicyclic graphs, and the corresponding extremal bicyclic graphs are also determined.

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Acknowledgements

The authors are much grateful to two anonymous referees for their valuable comments on our paper, which have considerably improved the presentation of this paper.

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Correspondence to Kinkar Chandra Das.

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Tingzeng Wu is supported by the National Natural Science Foundation of China (No. 11761056), the Natural Science Foundation of Qinghai Province (2016-ZJ-947Q), the Ministry of Education Chunhui Project (No. Z2017047) and the Key Project of QHMU (2019XJZ10). Kinkar Chandra Das is supported by the National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642.

Communicated by Maria Aguieiras de Freitas.

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Wu, T., Das, K.C. On the permanental sum of bicyclic graphs. Comp. Appl. Math. 39, 72 (2020). https://doi.org/10.1007/s40314-020-1108-x

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Keywords

  • Permanent
  • Permanental polynomial
  • Permanental sum
  • Bicyclic graph

Mathematics Subject Classification

  • MSC 05C31
  • MSC 05C75
  • MSC 15A15