Related papers: On the limit of large girth graph sequences

Motivated by the work of Lov\'asz and Szegedy on the convergence and limits of dense graph sequences, we investigate the convergence and limits of finite trees with respect to sampling in normalized distance. Based on separable real trees,…

Erd\H{o}s and Palka initiated the study of the maximal size of induced trees in random graphs in 1983. They proved that for every fixed $0<p<1$ the size of a largest induced tree in $G_{n,p}$ is concentrated around $2\log_q (np)$ with high…

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible…

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore…

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral…

We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree…

A multi-type branching process is defined as a random tree with labeled vertices, where each vertex produces offspring independently according to the same multivariate probability distribution. We demonstrate that in realizations of the…

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained…

We study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph (the spine) to which leaves are attached. We prove that the model has two phases, a generic phase where the spine becomes infinitely…

We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a…

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the…

It is conjectured that the Laplacian spectrum of a graph is majorized by its conjugate degree sequence. In this paper, we prove that this majorization holds for a class of graphs including trees. We also show that a generalization of this…

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting…

We prove that if a tree $T$ has $n$ vertices and maximum degree at most $\Delta$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,\Delta\log^5 n/n)$.

We study random digraphs on sequences of expanders with bounded average degree {which converge locally in probability}. We prove that the threshold for the existence of a giant strongly connected component, as well as the asymptotic…

We prove the existence of a limit of the finite volume probability measures generated by tree growth rules in Ford's alpha model of phylogenetic trees. The limiting measure is shown to be concentrated on the set of trees consisting of…

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$…

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric…

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by…

We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…