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Degree Kirchhoff Index of Bicyclic Graphs

Published online by Cambridge University Press:  20 November 2018

Zikai Tang  and
Hanyuan Deng
Zikai Tang
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P.R. China e-mail: [email protected] e-mail: [email protected]
Hanyuan Deng
Affiliation:
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P.R. China e-mail: [email protected] e-mail: [email protected]
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Abstract

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Let $G$ be a connected graph with vertex set $V\left( G \right)$.The degree Kirchhoff index of $G$ is defined as ${{S}^{\prime }}\left( G \right)\,=\,\sum{_{\left\{ u,v \right\}\,\subseteq \,V\left( G \right)}d\left( u \right)d\left( v \right)R\left( u,\,v \right)}$, where $d\left( u \right)$ is the degree of vertex $u$, and $R\left( u,\,v \right)$ denotes the resistance distance between vertices $u$ and $v$. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoff index among all $n$-vertex bicyclic graphs with exactly two cycles.

Keywords

05C1205C35degree Kirchhoff indexresistance distancebicyclic graphextremal graph
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 1 , 01 March 2017 , pp. 197 - 205
Copyright
Copyright © Canadian Mathematical Society 2017

References

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