  Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and
Das recently introduced the Steiner Gutman index by incorporating Steiner
distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index}
$\operatorname{SGut}_k(G)$ of $G$ is defined by $\operatorname{SGut}_k(G)$
$=\sum_{S\subseteq V(G), \ |S|=k}\left(\prod_{v\in S}deg_G(v)\right) d_G(S)$,
in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree
of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on
$\operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results
for the parameter $\operatorname{SGut}_k$. We obtain sharp upper and lower
bounds of $\operatorname{SGut}_k(G)+\operatorname{SGut}_k(\overline{G})$ and
$\operatorname{SGut}_k(G)\cdot \operatorname{SGut}_k(\overline{G})$ for a
connected graph $G$ of order $n$, $m$ edges and maximum degree $\Delta$,
minimum degree $\delta$.
