# Steiner diameter, maximum degree and size of a graph

## TL;DR

This paper investigates the minimum size of graphs with given order, Steiner diameter, and maximum degree, focusing on cases where the Steiner parameter k is close to the order n.

## Contribution

It extends previous work by analyzing the minimum size of graphs for Steiner diameter constraints when k is near n, specifically for n-3 ≤ k ≤ n.

## Key findings

- Characterization of graphs with minimal size for given Steiner diameter and maximum degree.
- Results for the cases when k equals n, n-1, n-2, and n-3.
- New bounds and structural properties of such graphs.

## 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 $n-3\leq k\leq n$.