  The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian
and Zou 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} $d(S)$ among the vertices of $S$ is the
minimum size among all connected subgraphs whose vertex sets contain $S$. Let
$n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner
$k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max
\{d(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S \}$. Furthermore, the
Steiner \emph{$k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in
V(G)\}$. In 2011, Chartrand, Okamoto, Zhang showed that $k-1\leq sdiam_k(G)\leq
n-1$. In this paper, graphs with $sdiam_k(G)=\ell$ for $k=n,n-1,n-2,n-3$ and
$k-1\leq \ell \leq n-1$ are characterized, respectively.
