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

Abstract Let G be a connected graph. The eccentric connectivity index ξ c ( G ) of G is defined as ξ c ( G ) = ∑ v ∈ V G d G ( v ) ϵ G ( v ) , where the eccentricity ϵ G ( v ) = max u ∈ V G 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 ξ 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 ξ c ( G ) , respectively. All the corresponding extremal graphs are identified.

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