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

Let Gbe a connected graph with vertex set V-G. The eccentric resistance-distance sum of Gis defined as xi(R)(G) = Sigma({u,v}) (subset of VG)(epsilon(G)(u) + epsilon(G)(v))R-uv, where epsilon(G)(center dot) is the eccentricity of the corresponding vertex and R-uv is the resistance-distance between uand vin 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 3having the smallest eccentric resistance-distance sum are identified. Furthermore, the n-vertex unicyclic graphs with given girth having the smallest and second smallest eccentric resistancedistance sums are identified, respectively. Consequently, n-vertex unicyclic graphs having the smallest and second smallest eccentric resistance-distance sums are characterized, respectively. (c) 2021 Elsevier Inc. All rights reserved.