Note on degree Kirchhoff index of graphs

Abstract

The degree Kirchhoff index of a connected graph $G$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $G$‎. ‎Bounds for the degree Kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎The special case of regular graphs is analyzed‎.