Steiner Harary Index

Yaping Mao

Kragujevac Journal of Mathematics, Tome 42 (2018) no. 1, p. 29 .

Résumé

The Harary index $H(G)$ of a connected graphs $G$ is defined as $H(G)=\sum_{u,v\in V(G)}\frac{1}{d_G(u,v)}$ 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 $2$ and $S\subseteq V(G)$, the { Steiner distance\/} $d_G(S)$ of the vertices of $S$ is the minimum size of all connected subgraphs whose vertex set contain $S$. Recently, Furtula, Gutman, and Katani-� introduced the concept of Steiner Harary index and give its chemical applications. The { $k$-center Steiner Harary index\/} $SH_k(G)$ of $G$ is defined by $SH_k(G)=\sum_{S\subseteq V(G), \ |S|=k}\frac{1}{d_G(S)}$. Expressions for $SH_k$ for some special graphs are obtained. We also give sharp upper and lower bounds of $SH_k$ of a connected graph, and establish some of its properties in the case of trees.

Détail

Classification : 05C12 05C05, 05C35
Keywords: Distance, Steiner distance, Harary index, Steiner Harary $k$-index