Some inequalities for the Kirchhoff index of graphs

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Authors :

Igor Milovanovic 1 *, Emina Milovanovic 1 , Marjan Matejic 1 and Edin Glogic 2

Author Address :

1 Faculty of Electronic Engineering, 18000 Nis, Serbia.
2 State University of Novi Pazar, 36300 Novi Pazar, Serbia.

*Corresponding author.

Abstract :

Let $G$ be a simple connected graph of order $n$, sequence of vertex degrees $d_1geq d_2geqcdotsgeq d_n>0$ and Laplacian eigenvalues $mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. With $Pi_1=Pi_1(G)=prod_{i=1}^n d_i^2$ we denote the multiplicative first Zagreb index of graph, and $Kf(G)=nsum_{i=1}^{n-1} frac{1}{mu_i}$ the Kirchhoff index of $G$. In this paper we determine several lower and upper bounds for $Kf$ depending on some of the graph parameters such as number of vertices, maximum degree, minimum degree, and number of spanning trees or multiplicative Zagreb index.

Keywords :

Kirchhoff index, Laplacian eigenvalues (of graph), vertex degree.



Article Info :

Received : November 24, 2017; Accepted : January 29, 2018.



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