On the minimum eccentric distance sum of bipartite graphs with some given parameters

The eccentric distance sum is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If G=(VG,EG)G=(VG,EG) is a simple connected graph, then the eccentric distance sum (EDS) of G   is defined as ξd(G)=∑v∈VGεG(v)DG(v)ξd(G)=∑v∈VGεG(v)DG(v), where εG(v)εG(v) is the eccentricity of the vertex v   and DG(v)=∑u∈VGdG(u,v)DG(v)=∑u∈VGdG(u,v) is the sum of all distances from the vertex v. Much extremal work has been done on trees with some given parameters by Yu et al. (2011) [25], Li et al. (2012) [19], and Geng et al. (2013) [6]. It is natural to consider this extremal problem on bipartite graphs with some given parameters. In this paper, sharp lower bound on the EDS in the class of all connected bipartite graphs with a given matching number q   is determined, the minimum EDS is realized only by the graph Kq,n−qKq,n−q. The extremal graph with the minimum EDS in the class of all the n-vertex connected bipartite graphs of odd diameter is characterized. All the extremal graphs having the minimum EDS in the class of all connected n-vertex bipartite graphs with a given vertex connectivity are identified as well.