Bicyclic graphs with extremal degree resistance distance

Abstract

Let $r(u,v)$ be the resistance distance between two vertices $u,v$ of a simple graph $G$, which is the effective resistance between the vertices in the corresponding electrical network constructed from $G$ by replacing each edge of $G$ with a unit resistor. The degree resistance distance of a simple graph $G$ is defined as $D_R(G)=\sum _{\{u,v\}\subseteq V(G)}[d(u)+d(v)]r(u,v),$ where $d(u)$ is the degree of the vertex $u$. In this paper, the bicyclic graphs with extremal degree resistance distance are strong-minded. We first determine the $n$-vertex bicyclic graphs having precisely two cycles with minimum and maximum degree resistance distance. We then completely characterize the bicyclic graphs with extremal degree resistance distance.