Ordering connected graphs having small degree distances

The concept of degree distance of a connected graph GG is a variation of the well-known Wiener index, in which the degrees of vertices are also involved. It is defined by D′(G)=∑x∈V(G)d(x)∑y∈V(G)d(x,y)D′(G)=∑x∈V(G)d(x)∑y∈V(G)d(x,y), where d(x)d(x) and d(x,y)d(x,y) are the degree of xx and the distance between xx and yy, respectively. In this paper it is proved that connected graphs of order n≥4n≥4 having the smallest degree distances are K1,n−1,BS(n−3,1)K1,n−1,BS(n−3,1) and K1,n−1+eK1,n−1+e (in this order), where BS(n−3,1)BS(n−3,1) denotes the bistar consisting of vertex disjoint stars K1,n−3K1,n−3 and K1,1K1,1 with central vertices joined by an edge.