Degree Kirchhoff Index of Bicyclic Graphs

Tang, Zikai ; Deng, Hanyuan

Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 197-205

Résumé

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.

Détail

DOI : 10.4153/CMB-2016-063-5

Mots-clés : 05C12, 05C35, degree Kirchhoff index, resistance distance, bicyclic graph, extremal graph