Abstract
The square of a graph \(G\), denoted by \(G^{2}\), is a graph with the same vertex set as \(G\), in which two vertices are adjacent if and only if their distance is at most \(2\) in \(G\). For \(S\subseteq V(G)\), the Steiner distance \(d(S)\) of \(S\)
is the minimum size of a connected subgraph of \(G\) whose vertex set contains \(S\). The \(k\)th Steiner Wiener index \(SW_{k}(G)\) of \(G\) is defined as the sum of Steiner distances of all \(k\)-element subsets of \(V(G)\). In this
paper, we show that for any tree \(T\) of order \(n\), \[ SW_3(S^2_n)\leq SW_3(T^2)\leq SW_3(P^2_n), \] where \(S_n\) and \(P_n\) are the star and path of the order \(n\), respectively. Let \(G\) be a connected graph of order \(n\geq 5\)
with connected complement \(\overline{G}\). We establish the Nordahaus-Gaddum type result for a connected graph \(G\) with connected complement \(\overline{G}\): \[ 4\binom{n}{3}\leq SW_3(G^2)+SW_3(\overline{G}^2)\leq SW_3(P^2_n)+SW_3(\overline{P_n}^2),
\] and \[ 4\leq sdiam_3(G^2)+sdiam_3(\overline{G}^2)\leq \left\{ \begin{aligned} &\left\lceil\frac{n}{2}\right\rceil+2 &\text{if \(n\geq 9\)}\\ &6 &\text{otherwise}, \end{aligned} \right. \] where \(sdiam_3(G)\) is Steiner \(3\)-diameter
of \(G\).
