Complete characterization of bicyclic graphs with the maximum and second-maximum degree Kirchhoff index

The degree Kirchhoff index (or multiplicative degree Kirchhoff index) of a connected simple graph G is defined as S′(G)=∑{u,v}⊆V(G)dG(u)dG(v)RG(u,v), where dG(u) is the degree of a vertex u in G and RG(u, v) is the resistance distance between the vertices u and v. In this paper, we completely characterize the bicyclic graphs of order n ≥ 6 having the maximum degree Kirchhoff index. Moreover, the bicyclic graphs of order n ≥ 7 with the second-maximum degree Kirchhoff index have also been determined.