Degree Distance and Minimum Degree

Abstract

Let G be a finite connected graph of order n, minimum degree δ and diameter d. The degree distance u, v), where deg w is the degree of vertex w and d(u, v) denotes the distance between u and v. In this paper, we find an asymptotically sharp upper bound on the degree distance in terms of order, minimum degree and diameter. In particular, we prove that

Extremal properties of distance-based graph invariants for $k$-trees

“…It was proposed by Dobrynin and Kochetova [11] and Gutman [15], independently. For more information on this graph invariant one may be referred to [1], [8], [32], [37] and their references.…”

The Vertex Degree Distances of One Vertex Union of the Cycle and the Star

“…Topological indices and graph invariants based on the distances between the vertices of a graph are widely used in theoretical chemistry to establish relations between the structure and the properties of molecules [11,12] . The Wiener index is a well-known topological index which equals the sum of distances between all pairs of vertices of a molecular graph [1,8,10,13] . The degree distance of a graph is a graph invariant that is more sensitive than the Wiener index [2][3][4][5][6] .…”

General degree distance of graphs

“…was introduced independently by Dobrynin and Kochetova [5] and Gutman [6]. Bounds on the degree distance for graphs with given vertex connectivity were obtained in [2], bounds for graphs with given edge connectivity in [1], bounds for graphs of minimum degree in [13], bounds for cacti in [20] and bounds for bicyclic graphs in [3]. The degree distance for unicyclic graphs with prescribed matching number was investigated in [10] and graph products were studied in [19] and [16].…”