The Diameter of Dense Random Regular Graphs

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(Gn;d) of a random d-regular graph G(n,d) of order n is known to be asymptotically "optimal" as n -> infinity. Bollobas and de la Vega (1982) proved that diam(G(n,d)) = (1 + o(1)) D-0(n, d) = (1 + o(1)) log(d-1) n holds w.h.p. (with high probability) for fixed d >= 3, whereas there exists a gap diam(G(n,d)) - D-0(n, d) = Omega (log log n). In this paper, we investigate the gap diam(G(n,d)) - D-0(n, d) for d = (beta+o(1)) n(alpha) where alpha is an element of (0, 1) and beta > 0 are arbitrary constants. We prove that diam(G(n,d)) = left perpendicular alpha(-1)right perpendicular + 1 holds w.h.p. for such d. Our result implies that the gap is 1 if alpha(-1) is an integer and d >= n(alpha), and is 0 otherwise. One can easily obtain that diam(Gn,d) <= left perpendicular alpha(-1)right perpendicular + 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)) >= left perpendicular alpha(-1)right perpendicular + 1 holds w.h.p. by the analysis of the distances of fixed vertex pairs.