Nordhaus-Gaddum Type results for the Harary index of graphs

The \emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=\sum {u,v\in V(G)}\frac{1}{d_G(u,v)}$ 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 \emph{Steiner distance} $d_G(S)$ of the vertices of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. Recently, Furtula, Gutman, and Katani'{c} introduced the concept of Steiner Harary index and gave its chemical applications. The \emph{$k$-center Steiner Harary index} $SH_k(G)$ of $G$ is defined by $SH_k(G)=\sum_{S\subseteq V(G),|S|=k}\frac{1}{d_G(S)}$. In this paper, we get the sharp upper and lower bounds for $SH_k(G)_k(\overline{G})$ and $SH_k(G)\cdot SH_k(\overline{G})$, valid for any connected graph $G$ whose complement $\overline {G}$ is also connected.