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

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.09424
 Bibcode:
 2019arXiv190509424Z
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Combinatorics;
 05C50;
 05C90;
 G.2
 EPrint:
 In this paper, we firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of On. Further, the formulas for those two resistance distancebased graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degreeKirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph