Extremal bipartite graphs and unicyclic graphs with respect to the eccentric resistance-distance sum

Let G be a connected graph with vertex set VG. The eccentric resistance-distance sum of G is defined as ξR(G)=∑{u,v}⊆VG(εG(u)+εG(v))Ruv, where εG(⋅) is the eccentricity of the corresponding vertex and Ruv is the resistance-distance between u and v in G. In this paper, among the bipartite graphs of diameter 2, the graphs having the smallest and the largest eccentric resistance-distance sums are characterized, respectively. Among the bipartite graphs of diameter 3, the graphs having the smallest and second smallest eccentric resistance-distance sums are characterized, respectively. As well the graphs of diameter 3 having the smallest eccentric resistance-distance sum are identified. Furthermore, the n-vertex unicyclic graphs with given girth having the smallest and second smallest eccentric resistance-distance sums are identified, respectively. Consequently, n-vertex unicyclic graphs having the smallest and second smallest eccentric resistance-distance sums are characterized, respectively.