Abstract

Let G be a connected graph. The eccentric connectivity index ξc(G) of G is defined as ξc(G)=∑v∈VGdG(v)ϵG(v), where the eccentricity ϵG(v)=maxu∈VGdG(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 ξc(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 ξc(G), respectively. All the corresponding extremal graphs are identified.

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