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 ℓ was first introduced by Erdös and Rényi. 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 ℓ, where 3 ≤ k ≤ n, and studied this new problem when k = 3. In this paper, we investigate the problem when n − 3 ≤ k ≤ n.

Steiner Distance in Graphs--A Survey

“…For 2 ≤ ℓ ≤ n, e n (n, ℓ, n − 1) = n − 1.For k = n − 1 and k = n − 2, Mao and Wang[101] derived the following results.Proposition 9.4 [101] (1) For 2 ≤ ℓ ≤ n − 1, e n−1 (n, ℓ, n − 1) = n − 1. For 2 ≤ ℓ ≤ n − 1, e n−1 (n, ℓ, n − 2) = n + ℓ − 2.…”