Some extremal properties of the degree distance of a graph (Q1961241)

For a vertex \(x\) of a graph \(G\) let \(D(x)\) denote the sum of all distances \(d(x,y)\) where \(y\) ranges over the vertex set \(V(G)\) of \(G\) and \(\deg(x)\) denote the degree of \(x\). The value \(D'(x):=\deg(x)D(x)\) is known as the degree distance of \(x\) and \(D'(G):=\sum_{x\in V(G)}D'(x)\) is the degree distance of \(G\). This paper deals with two conjectures of \textit{A. A. Dobrynin} and \textit{A. A. Kochetova} [Degree distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34, 1082-1086 (1994)] concerning the minimum and maximum values of \(D'(G)\) for graphs \(G\) of order \(n\). One conjecture is settled in the affirmative and the other is disproved. Exact values are given for trees.