On the minimal eccentric connectivity indices of bipartite graphs with some given parameters

Let G be a connected graph. The eccentric connectivity index xi(c)(G) of G is defined as xi(c)(G) = Sigma(v is an element of VG) d(G)(v)is an element of(G)(v), where the eccentricity epsilon(G)(v) = max(u)(is an element of VG) d(G)(v, u). Zhang et al. (2012) studied the minimal eccentric connectivity indices of graphs. As a continuance of it, in this paper we consider these problems on bipartite graphs. We obtain lower bounds on xi(c)(G) in terms of the number of edges among n-vertex connected bipartite graphs with given diameter. Among all connected bipartite graphs on n vertices with m edges and diameter at least s, and connected bipartite graphs on n vertices with diameter at least s, we establish the lower bounds on xi(c)(G), respectively. All the corresponding extremal graphs are identified. (C) 2018 Elsevier B.V. All rights reserved.