Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs
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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.
KeywordsLaplacian Normalized Laplacian Kirchhoff index Multiplicative degree-Kirchhoff index Spanning tree
Mathematics Subject Classification05C50 05C90
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.
- 4.Pan, Y., Li, J.: Graphs that minimizing symmetric division degree index. MATCH Commun. Math. Comput. Chem. 82, 43–55 (2019)Google Scholar
- 9.Klein, D.J.: Resistance–distance sum rules. Croat. Chem. Acta 75, 633–649 (2002)Google Scholar
- 15.Mohar, B.: The Laplacian spectrum of graphs. In: Alavi, Y., Chartrand, G., Oellermann, O.R., Schwenk, A.J. (eds.) Graph Theory, Combinatorics, and Applications, vol. 2, pp. 871–898. Wiley, Hoboken (1991)Google Scholar