Steiner 3-diameter, maximum degree and size of a graph

Abstract

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. 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. In this paper, we generalize the above problem for Steiner k-diameter, and study the problem of determining the minimum size of a graph of order n whose Steiner 3-diameter is at most d and whose maximum is at most ℓ.

Steiner Distance in Graphs--A Survey

“…Mao [96] The following results can be easily proved. (2) For ℓ = n − 2, e 3 (n, ℓ, 2) = n 2 − n 2 for n even; e 3 (n, ℓ, 2) = ∞ for n odd.…”

“…For n ≥ 6, e 3 (n, 5, n − 4) = n = 6, n + 2 if n = 7, n − 1 if n ≥ 8. For n ≥ 7 and 6 ≤ ℓ ≤ n − 1, e 3 (n, ℓ, n − 4) = n − 1.Mao[96] also constructed a graph and gave an upper bound of e 3 (n, ℓ, d) for general ℓ and d.Proposition 9.2 For 4 ≤ d ≤ n − 1 and 2 ≤ ℓ ≤ n − 1, e 3 (n, ℓ, d) ≤ (n − d + 1)(n − d + 2) 2 + d − 3.…”

Steiner diameter, maximum degree and size of a graph

“…Mao [37] considered the generalization of the above problem. Let d, ℓ and n be natural numbers, d < n and ℓ < n. Denote by H k (n, ℓ, d) the set of all graphs of order n with maximum degree ℓ and…”

“…In [37], Mao focused their attention on the case k = 3, and studied the exact value of e 3 (n, ℓ, d)…”