The Diameter of Dense Random Regular Graphs

Abstract

There is a tight upper bound on the order (the number of vertices) of any d-regular graph of diameter D, known as the Moore bound in graph theory. This bound implies a lower bound D 0 (n, d) on the diameter of any d-regular graph of order n. Actually, the diameter diam (G n,d ) of a random dregular graph G n,d of order n is known to be asymptotically "optimal" as n → ∞. Bollobás and de la Vega (1982) holds w.h.p. (with high probability) for fixed d ≥ 3, whereas there exists a gapIn this paper, we investigate the gap diam( (1)) n α where α ∈ (0, 1) and β > 0 are arbitrary constants. We prove that diam(G n,d ) = ⌊α −1 ⌋ + 1 holds w.h.p. for such d. Our result implies that the gap is 1 if α −1 is an integer and d ≥ n α , and is 0 otherwise. One can easily obtain that diam (G n,d ) ≤ ⌊α −1 ⌋ + 1 holds w.h.p. by using the embedding theorem due to Dudek et al. (2017). Our critical contribution is to show that diam (G n,d ) ≥ ⌊α −1 ⌋ + 1 holds w.h.p. by the analysis of the distances of fixed vertex pairs.

The Average Distance and the Diameter of Dense Random Regular Graphs

“…In the conference version of this paper[29], we proved Theorem 2.the electronic journal of combinatorics 27(3) (2020), #P3.62…”

Sandwiching random regular graphs between binomial random graphs

“…Remark 2.12. A recent paper by Shimizu [34] determined the diameter of G(n, d) for d ∼ βn α where β and α are positive constants. Our Theorem 2.11 recovers this result except when 1/α is an integer, in which case Theorem 2.11(b) yields a 2-point concentration.…”