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JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2019 Volume 16, Pages 1654–1661 (Mi semr1158)

This article is cited in 2 papers

Discrete mathematics and mathematical cybernetics

Elementary formulas for Kirchhoff index of Möbius ladder and Prism graphs

G. A. Baigonakovaa, A. D. Mednykhbc

a Gorno-Altaysk State University, 34, Socialisticheskaya str., Gorno-Altaysk, 639000, Russia
b Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia
c Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia

Abstract: Let $G$ be a finite connected graph on $n$ vertices with Laplacian spectrum $0=\lambda_1<\lambda_2\le\ldots\le\lambda_n.$ The Kirchhoff index of $G$ is defined by the formula
$$Kf(G)=n\sum\limits_{j=2}^n\frac{1}{\lambda_j}.$$
The aim of this paper is to find an explicit analytical formula for the Kirchhoff index of Möbius ladder graph $M_n=C_{2n}(1,n)$ and Prism graph $Pr_n=C_n\times P_2$. The obtained formulas provide a simple asymptotical behavior of both invariants as $n$ is going to the infinity.

Keywords: Laplacian matrix, circulant graph, Kirchhoff index, Wiener index, Chebyshev polynomial.

UDC: 519.175.3, 519.172

MSC: 05C30, 39A10

Received March 15, 2019, published November 21, 2019

Language: English

DOI: 10.33048/semi.2019.16.117



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