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 $2$ and $ssubseteq v(g)$, 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 $sh_k(g)=sum_{ssubseteq v(g),|s|=k}frac{1}{d_g(s)}$. in this paper, we get the sharp upper and lower bounds for $sh_k(g)+sh_k(overline{g})$ and $sh_k(g)cdot sh_k(overline{g})$, valid for any connected graph $g$ whose complement $overline {g}$ is also connected.