Steiner $4$-Diameter, Maximum Degree and Size of a Graph - Li, He; Zhang, Shumin; Zhu, Bo; Ye, Chengfu (Discontinuity, Nonlinearity, and Complexity, 2026)
The Steiner $k$-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. Let $d,\ell$ and $n$ be natural numbers and $d\lt n$, $\ell\lt n$. Denote by $\mathscr{H}_k(n,\ell,d)$ the set of all graphs of order $n$ with maximum degree $\ell$ and $sdiam_k(G)≤ d$. Let $e_k(n,\ell,d)=\min\{e(G):G\in \mathscr{H}_k(n,\ell,d)\}$. In this paper, we focus on the case $k=4$, and study the exact value of $e_4(n,\ell,d)$.