Graph limits of random graphs from a subset of connected k‐trees

Abstract

For any set Ω of non‐negative integers such that 0 1 , we consider a random Ω‐ k ‐tree G n , k that is uniformly selected from all connected k ‐trees of ( n + k ) vertices such that the number of ( k + 1)‐cliques that contain any fixed k ‐clique belongs to Ω. We prove that G n , k , scaled by 2 where H k is the k th harmonic number and σ Ω > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐ k ‐tree to an infinite but locally finite random Ω‐ k ‐tree G ∞, k .

Limits of random tree-like discrete structures

“…We apply results for extended fringe subtrees of random enriched trees to provide a Benjamini-Schramm limit of random k-trees. Even more ambitiously, we verify total variational convergence of o( √ n)-neighbourhoods, which is the strongest possible form of convergence in this context, as the diameter of random k-trees has order √ n [47]. We compare the limit graph with a local limit established in [47] that encodes convergence of neighbourhoods around a random k-clique.…”

“…Even more ambitiously, we verify total variational convergence of o( √ n)-neighbourhoods, which is the strongest possible form of convergence in this context, as the diameter of random k-trees has order √ n [47]. We compare the limit graph with a local limit established in [47] that encodes convergence of neighbourhoods around a random k-clique. The limit objects are distinct, which is already evident from the different behaviour of the degree of a random vertex and a vertex of a random front.…”

“…-We are interested in obtaining the Benjamini-Schramm limit of the random k-dimensional tree K n with n hedra as n becomes large. A local limit around (a fixed vertex of) a uniformly at random drawn front was established in [47]. As we shall see, the two limits are distinct, which is already evident from the fact that the degrees of the root vertices tends to different limit distributions.…”

Graph limits of random unlabelled k-trees

“…The Rayleigh distribution has been observed to arise as scaling limit of the distance of independent random vertices in random labelled k-trees by Darrasse and Soria [6], but the scaling constant of (2.10) differs from the labelled case. Drmota and both authors [8] gave a scaling limit for random labelled k-trees, of course also with a different scaling constant.…”

“…Drmota and the first author [7] established the following result, which shows the dominating role of the cycle type 1 k in this context. The class of labelled k-trees admits a recursive decomposition [8] that is based on k-trees rooted at a front of distinguishable vertices. Two such elements are considered isomorphic if there is a graph isomorphism that pointwise preserves the root-front.…”