Nordhaus-Gaddum-type inequality for the hyper-Wiener index of graphs when decomposing into three parts

Nordhaus-Gaddum-Type Theorem for Diameter of Graphs when Decomposing into Many Parts

Nordhaus-Gaddum type results for the Harary index of graphs

The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...

The Wiener number of powers of the Mycielskian

The Wiener number of a graph G is defined as 1 2 ∑ u,v∈V (G) d(u, v), d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, W (μ(S n)) ≤ W (μ(T k n )) ≤ W (μ(P k n )), where Sn, Tn and Pn denote a star, a gen...

The Nordhaus-Gaddum-type inequalities for the Zagreb index and co-index of graphs

Let k ≥ 2 be an integer, a k-decomposition (G1,G2, . . . ,Gk) of the complete graph Kn is a partition of its edge set to form k spanning subgraphs G1,G2, . . . ,Gk. In this contribution, we investigate the Nordhaus–Gaddum-type inequality of a k-decomposition of Kn for the general Zagreb index and a 2-decomposition for the Zagreb co-indices, respectively. The corresponding extremal graphs are ch...

The Wiener index of the kth power of a graph

The kth power of a graph G, denoted by Gk , is a graph with the same vertex set as G such that two vertices are adjacent in Gk if and only if their distance is at most k in G. The Wiener index is a distance-based topological index defined as the sum of distances between all pairs of vertices in a graph. In this note, we give the bounds on the Wiener index of the graph Gk . The Nordhaus–Gaddum-t...