Some Bounds for the Steiner-Harary Index of a Graph

Abstract

The Steiner distance for the set $S ⊆ V (G)$ would simply be the number of edges in the minimal subtree connecting them and is denoted as $d_G(S).$ The Steiner-Harary index is $SH_k(G),$ defined as the sum of the reciprocal of the Steiner distance for all subsets with $k$ vertices in $G.$ In this article, we calculate the exact value of $SH_k(G)$ for specific graphs and establish new best possible lower and upper bounds and characterization. Furthermore, we explore the relationship between $SH_k(G)$ and other graph indices based on Steiner distance.