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 (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.