Asymptotic enumeration of digraphs and bipartite graphs by degree sequence

Abstract

We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in-and out-degree sequences, for a wide range of parameters including the biregular case. In particular, for bipartite graphs with part sizes that are not highly unequal, our results cover an existing gap in known results between the sparse and dense cases that were proved by Wang in 2006 andby Canfield, Greenhill, andMcKay in 2008, respectively. For the range of parameters which our results cover, they imply that the degree sequence of a random bipartite graph with m edges and given part sizes is accurately modeled by a sequence of independent binomial random variables, conditional upon the sum of variables in each part being equal to m. This extends the known behavior in the sparse and dense cases to essentially the full spectrum of possible densities. A similar model also holds for loopless digraphs.

The spectral gap of random regular graphs

“…Our notation is similar to that of [14], but not completely consistent with it. We also denote , the “projection” of to $$$ v $$$.…”

“…The following recursive formulae for P d, (ab), Y d, (abc) are proven by Liebenau and Wormald in[14]:Proposition 2.8. If P d, (ab) > 0, then P d, (ab) = d(b) ( ∑ c∈ * (b) d(c) − B d−e a −e b , (ca) d(a) − B d−e b −e c , (ac) ⋅ 1 − P d−e b −e c , (bc) 1 − P d−e a −e b , (ab) ) −(2.5) where B d, (ab) is defined by B d, (ab) = ∑ c∈(a)⧵(b) P d, (ac) + ∑ c∈(a)∩(b) Y d, (acb).…”

Subgraph distributions in dense random regular graphs

“…In fact, certain comparisons to |G(n, d)| simplify the situation, allowing us to avoid repeating a careful saddle point analysis as in the work of McKay and Wormald [MW90]. The nontrivial expectation contributions E H when H is an even cycle come into play due to counting certain even-power monomials in a polynomial expansion associated to the edges of H. We further believe combining the general method of considering graph factors along with recent work of Liebenau and Wormald [LW17], which enumerates graphs of degrees of intermediate sparsity, can likely be used to address asymptotic distribution for regular graphs of all sparsities, a direction we plan to pursue in future work. Finally, we note that while Theorem 1.2 can handle subgraph counts of mildly growing size, the understanding of the asymptotic distributions of spanning subgraphs in dense random regular graphs is also of interest.…”

“…Let denote a uniformly random -regular graph. Note that unlike or , the edges in exhibit strong and non-obvious correlations and therefore even the question of determining the number of -regular graphs has a rich history drawing on techniques ranging from switchings developed by McKay [McK85] (and refined by McKay and Wormald [MW91]), a complex-analytic technique of McKay and Wormald [MW90], and recent breakthroughs using fixed-point iteration due to Liebenau and Wormald [LW17]. We refer the reader to the excellent survey of Wormald [Wor18] where the extensive history of this problem and various related enumeration problems are discussed.…”

Estimating contact network properties by integrating multiple data sources associated with infectious diseases

“…We calculate the number of networks in each of the 11 congruence classes by enumerating all 64 networks in and summarizing their degree distribution. For larger networks, graph enumeration methods exist for estimating the size of congruence classes 40,41 …”