Given a collection of n rooted trees with depth h, we give a necessary and sufficient condition for this collection to be the collection of h-depth universal covering neighborhoods at each vertex.
For @math , the number of labeled graphs with a given degree sequence is asymptotically known in many asymptotic regimes, see notably @cite_6 , [Theorem 2.16] MR1864966 and references therein. For general @math , this question has been recently adressed in @cite_2 in the regime where the maximal degree is uniformly bounded. The motivation came from the Benjamini-Schramm topology of rooted graphs.
@cite_6 @cite_2
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We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d1,…,dn). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0-1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 It is shown that almost all labelled r -regular graphs are r -connected, for any fixed r ≥3. This is an instance of a more general result which applies to labelled locally restricted graphs whose points have degrees lying between r and R , where r and R are fixed, 3≤ r ≤ R . The proof employs a known asymptotic formula for the numbers of these locally restricted graphs. It is also demonstrated that for fixed k >0 and r ≥3, almost all labelled r -regular graphs with girth at least [ k ( r −2)] are cyclically k -connected. This provides an asymptotic formula for the number of labelled cyclically k -connected r -regular graphs with p points, for fixed k and r . We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|1 3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|1 4). In particular, ifk=o(n1 2), the number of regular graphs of degreek and ordern is asymptotically @math Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences. Let Δ and n be natural numbers such that Δn = 2m is even and Δ ⩽ (2 log n )1 2 - 1. Then as n →, the number of labelled Δ-regular graphs on n vertices is asymptotic to e − λ − λ 2 ( 2 m ) ! m ! 2 m ( Δ ! ) m where λ = (Δ -1) 2. As a consequence of the method we determine the asymptotic distribution of the number of short cycles in graphs with a given degree sequence, and give analogous formulae for hypergraphs.