Steiner diameter, maximum degree and size of a graph

Abstract: The Steiner diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erd\"{o}s and R\'{e}nyi. Recently, Mao considered the problem of determining the minimum size of a graph of order $n$ whose Steiner $k$-diameter is at most $d$ and whose maximum is at most $\ell$, where $3\leq k\leq n$, and studied this new problem when $k=3$. In this paper, we investigate the problem when $n-3\leq k\leq n$.