Bicyclic graphs with maximum degree resistance distance

Abstract

Graph invariants, based on the distances between the
vertices of a graph, are widely used in theoretical chemistry.
Recently, Gutman, Feng and Yu ({\it Transactions on Combinatorics},
01 (2012) 27-40) introduced the {\it degree resistance distance} of
a graph $G$, which is defined as $D_{R}(G)=\sum_{\{u,v\}\subseteq
V(G)}[d_G(u)+d_G(v)]R_G(u,v)$, where $d_G(u)$ is the degree of
vertex $u$ of the graph $G$, and $R_G(u,v)$ denotes the resistance
distance between the vertices $u$ and $v$ of the graph $G$. Further,
they characterized $n$-vertex unicyclic graphs having minimum and
second minimum degree resistance distance. In this paper, we
characterize $n$-vertex bicyclic graphs
having maximum degree resistance distance.