Degree distance of unicyclic and bicyclic graphs

Let GG be a connected graph with vertex set V(G)V(G). The degree distance of GG is defined as D′(G)=∑{u,v}⊆V(G)(degG(u)+degG(v))d(u,v), where degG(u) is the degree of vertex uu, and d(u,v)d(u,v) denotes the distance between uu and vv. Here we characterize nn-vertex unicyclic graphs with girth kk, having minimum and maximum degree distance, respectively. Furthermore, we prove that the graph BnBn, obtained from two triangles linked by a path, is the unique graph having the maximum degree distance among bicyclic graphs, which resolves a recent conjecture of Tomescu.