Resistance distance-based 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 O n be a linear crossed octagonal graph. Recently, Pan and Li (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.

Spectral techniques and mathematical aspects of K 4 chain graph

“…The expressions for number of spanning trees and Kirchhoff index for different networks, including cycles [23], linear polyomino chains [24], linear crossed octagonal chains [25], circulant graphs [26] , linear crossed hexagonal chains [27], cylinder phenylene chains [28], linear hexagonal chains [29], ladder graphs [30], ladderlike chains [31], and linear phenylenes [32] were calculated by different researchers. For more related results, the readers can see [33][34][35][36][37][38].…”

Computing the Normalized Laplacian Spectrum and Spanning Tree of the Strong Prism of Octagonal Network

“…Since the Kirchhoff index and multiplicative degree-Kirchhoff index have been widely used in the domains of physics, chemistry, and network science. During the previous few decades, many scientists have been working on explicit formulae for the Kirchhoff and degree-Kirchhoff indices of graphs with particular structures, such as cycles [18], complete multipartite graphs [19], generalized phenylene [20], crossed octagonal [21], hexagonal chains [22], pentagonal-quadrilateral network [23], and so on. Other research on the Kirchhoff index and the multiplicative degree-Kirchhoff index of a graph has been published (see [24][25][26][27][28][29][30][31]).…”

The (multiplicative degree-)Kirchhoff index of graphs derived from the Catersian product of $S_n$ and $K_2$

“…linear crossed hexagonal chain [32], Möbius hexagonal chain [33], and periodic linear chains [34], linear octagonal chain [35], linear octagonal-quadrilateral chain [36], and linear crossed octagonal chain [37].…”