The concept of degree distance of a connected graph G 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), where d (x) and d (x, y) are the degree of x and the
distance between x and y, respectively. In this paper it is proved that connected graphs of
order n≥ 4 having the smallest degree distances are K1, n− 1, BS (n− 3, 1) and K1, n− 1+ e
(in this order), where BS (n− 3, 1) denotes the bistar consisting of vertex disjoint stars K1, n …

The parameter D′(G) of a connected graph G is called the degree distance of G and was
introduced by Dobrynin and Kochetova and independently by Gutman as a weighted
version of the Wiener index. It is defined by D′(G)=∑ x∈ V (G) d (x)∑ y∈ V (G) d (x, y),
where d (x) and d (x, y) are the degree of x and the distance between x and y, respectively.
In a previous paper [14] the first author found three graphs having smallest degree
distances. Here the next six graphs of order n in this sequence are determined provided n≥ …