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Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs

  • Jing Zhao
  • Jia-Bao LiuEmail author
  • Sakander Hayat
Original Research
  • 32 Downloads

Abstract

Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let \(O_n\) be a linear crossed octagonal graph. Recently, Pan and Li (Int J Quantum Chem 118(24):e25787, 2018) derived the closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of \(H_n\). They pointed that it is interesting to give the explicit formulas for the Kirchhoff and multiplicative degree-Kirchhoff indices of \(O_n\). Inspired by these, in this paper, two resistance distance-based graph invariants, namely, Kirchhoff and multiplicative degree-Kirchhoff indices are studied. We firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of \(O_n\). Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph.

Keywords

Laplacian Normalized Laplacian Kirchhoff index Multiplicative degree-Kirchhoff index Spanning tree 

Mathematics Subject Classification

05C50 05C90 

Notes

Acknowledgements

This work is supported by National Natural Science Foundation of China (Nos. 11601006, 11801007), China Postdoctoral Science Foundation under Grant 2017M621579; the Postdoctoral Science Foundation of Jiangsu Province under Grant 1701081B; Project of Anhui Jianzhu University under Grant no. 2016QD116 and 2017dc03, Natural Science Foundation of Anhui Province, 1808085MA17.

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Copyright information

© Korean Society for Informatics and Computational Applied Mathematics 2019

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsAnhui Jianzhu UniversityHefeiPeople’s Republic of China
  2. 2.School of MathematicsSoutheast UniversityNanjingPeople’s Republic of China
  3. 3.Faculty of Engineering SciencesGIK Institute of Engineering Sciences and TechnologyTopiPakistan

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