The minimal degree Kirchhoff index of bicyclic graphs

The degree Kirchhoff index of graph $G$ is defined as $K{f}^{\ast }(G)=\sum _{u,v\subseteq V(G)}d(u)d(v)r_G(u,v)$, where $d(u)$ is the degree of vertex $u$ and $r_G(u,v)$ is the resistance distance between the vertices $u$ and $v$. In this paper, we characterize bicyclic graphs with exactly two cycles having the minimum degree Kirchhoff index of order $n\ge 5$. Moreover, we obtain the minimum degree Kirchhoff index on bicyclic graphs of order $n\ge 4$ with exactly three cycles, and all bicyclic graphs of order $n\ge 4$ where the minimum degree Kirchhoff index has been obtained.