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.
@cite_6: Consider a random multigraph G* with given vertex degrees d1,…,dn, constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges , the probability that the multigraph is simple stays away from 0 if and only if . This was previously known only under extra assumptions on the maximum degree maxidi. We also give an asymptotic formula for this probability, extending previous results by several authors.
@cite_2: Consider the Erdős–Renyi random graph on (n ) vertices where each edge is present independently with probability ( n ), with ( >0 ) fixed. For large (n ), a typical random graph locally behaves like a Galton–Watson tree with Poisson offspring distribution with mean ( ). Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform random graphs with a given neighborhood distribution, provided the latter is supported on trees. We also introduce a new class of unimodular random trees, which generalizes the usual Galton Watson tree with given degree distribution to the case of neighborhoods of arbitrary finite depth. These generalized Galton Watson trees turn out to be useful in the analysis of unimodular random trees and may be considered to be of interest in their own right.
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.