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SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

Part of: Graph theory

Published online by Cambridge University Press:  04 October 2017

I. MILOVANOVIĆ Open the ORCID record for I. MILOVANOVIĆ [Opens in a new window] ,
M. MATEJIĆ ,
E. GLOGIĆ  and
E. MILOVANOVIĆ
I. MILOVANOVIĆ*
Affiliation:
Faculty of Electronic Engineering, Niš, Serbia email igor@elfak.ni.ac.rs
M. MATEJIĆ
Affiliation:
Faculty of Electronic Engineering, Niš, Serbia email marjan.matejic@elfak.ni.ac.rs
E. GLOGIĆ
Affiliation:
State University of Novi Pazar, Novi Pazar, Serbia email edin_gj@hotmail.com
E. MILOVANOVIĆ
Affiliation:
Faculty of Electronic Engineering, Niš, Serbia email ema@elfak.ni.ac.rs
*
✉igor@elfak.ni.ac.rs
Rights & Permissions [Opens in a new window]

Abstract

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Let $G$ be a simple connected graph with $n$ vertices and $m$ edges and $d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$ its sequence of vertex degrees. If $\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$ are the Laplacian eigenvalues of $G$, then the Kirchhoff index of $G$ is $\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$. We prove some new lower bounds for $\mathit{Kf}(G)$ in terms of some of the parameters $\unicode[STIX]{x1D6E5}=d_{1}$, $\unicode[STIX]{x1D6E5}_{2}=d_{2}$, $\unicode[STIX]{x1D6E5}_{3}=d_{3}$, $\unicode[STIX]{x1D6FF}=d_{n}$, $\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$ and the topological index $\mathit{NK}=\prod _{i=1}^{n}d_{i}$.

Keywords

Kirchhoff indexLaplacian eigenvalues (of a graph)vertex degree

MSC classification

Primary: 05C12: Distance in graphs
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 1 , February 2018 , pp. 1 - 10
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the Serbian Ministry for Education, Science and Technological Development.

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