In 1994, Dobrynin and Kochetova introduced the concept of degree distance DD(Γ) a connected graph Γ. Let dΓ(S) be Steiner k-distance S⊆V(Γ). The Wiener k-index or k-center indexSWk(Γ) Γ is defined by SWk(Γ)=∑|S|=kS⊆V(Γ)dΓ(S). distanceSDDk(Γ) SDDk(Γ)=∑|S|=kS⊆V(Γ)∑v∈SdegΓ(v)dΓ(S), where degΓ(v) vertex v in this paper, we consider Nordhaus–Gaddum-type results for SWk(Γ) SDDk(Γ). Upper bounds on SW...

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanne...

The Harary index is defined as the sum of reciprocals distances between all pairs vertices a connected graph G = (V, E). In this paper we introduce Index Power 3 Tree Mean graphs.

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let G (n, r) be the set of cacti of order n and with r cycles, ξ(2n, r) the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we give the sharp upper bounds of t...

For a connected graph G and an non-empty set S ⊆ V (G), the Steiner distance dG(S) among the vertices of S is defined as the minimum size among all connected subgraphs whose vertex sets contain S. This concept represents a natural generalization of the concept of classical graph distance. Recently, the Steiner Wiener index of a graph was introduced by replacing the classical graph distance used...

In this paper we answer a question posed [1]. We show that χ(H2m,3m+2) = m + k + 1 such that k = m/2 and H2m,3m+2 is the Harary graph. Mathematics Subject Classification: 05C15