On the Number of Spanning Trees in Random Regular Graphs

Abstract

Let d ≥ 3 be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random d-regular graph with n vertices. (The asymptotics are as n → ∞, restricted to even n if d is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) d. Numerical evidence is presented which supports our conjecture.

Spanning trees in random regular uniform hypergraphs

“…Greenhill, Kwan and Wind [18] determined asymptotic distribution of spanning trees for random cubic graphs, that is, when (r, s) = (3,2). The aim of this paper is to generalise the results of [18] to random r-regular s-uniform hypergraphs for any fixed r, s 2 with (r, s) = (2, 2).…”

“…Greenhill, Kwan and Wind [18] determined asymptotic distribution of spanning trees for random cubic graphs, that is, when (r, s) = (3,2). The aim of this paper is to generalise the results of [18] to random r-regular s-uniform hypergraphs for any fixed r, s 2 with (r, s) = (2, 2). As noted in [18], for the case where (r, s) = (2, 2), a 2-regular graph has a spanning tree if and only if it is connected (that is, forms a Hamilton cycle).…”

“…The aim of this paper is to generalise the results of [18] to random r-regular s-uniform hypergraphs for any fixed r, s 2 with (r, s) = (2, 2). As noted in [18], for the case where (r, s) = (2, 2), a 2-regular graph has a spanning tree if and only if it is connected (that is, forms a Hamilton cycle). Thus it follows from [35,Equation (11)] that as n → ∞, Pr(G n,2,2 contains a spanning tree) ∼ 1 2 e 3/4 π n → 0.…”

“…(In fact a more general result is proved in [2], which covers irregular degree sequences and allows s and the maximum degree to grow slowly with n. ) In the graph case, the asymptotic formula for EY G was known up to a constant factor by the results of McKay [23] (who also considered irregular, slowly-growing degrees), and then this constant factor was calculated precisely in [18,Theorem 1.1].…”

“…The only previously-known case of Theorem 1.2 was (r, s) = (3, 2) (cubic graphs), proved by Greenhill, Kwan and Wind [18,Theorem 1.2]. The authors of [18] also conjectured an expression for the asymptotic distribution of Y G when s = 2 and r 4. Substituting s = 2 into Theorem 1.2 verifies that their conjecture is true.…”