65,349 research outputs found
On the order of the largest induced tree in a random graph
AbstractConsider a random graph K(n, p) with n labeled vertices in which the edges are chosen independently and with a probability p. Let Tn(p) be the order of the largest induced tree in K(n, p). Among other results it is shown, using an algorithmic approach, that if p=(c log n)/n, where c ≥ e is a constant, then for any fixed ε > 01c−εlog lognlognn<Tn(p)<2c+εlog lognlogn almost surely
On the structure of random graphs with constant -balls
We continue the study of the properties of graphs in which the ball of radius
around each vertex induces a graph isomorphic to the ball of radius in
some fixed vertex-transitive graph , for various choices of and .
This is a natural extension of the study of regular graphs. More precisely, if
is a vertex-transitive graph and , we say a graph is
{\em -locally } if the ball of radius around each vertex of
induces a graph isomorphic to the graph induced by the ball of radius
around any vertex of . We consider the following random graph model: for
each , we let be a graph chosen uniformly at
random from the set of all unlabelled, -vertex graphs that are -locally
. We investigate the properties possessed by the random graph with
high probability, for various natural choices of and .
We prove that if is a Cayley graph of a torsion-free group of polynomial
growth, and is sufficiently large depending on , then the random graph
has largest component of order at most with high
probability, and has at least automorphisms with high
probability, where depends upon alone. Both properties are in
stark contrast to random -regular graphs, which correspond to the case where
is the infinite -regular tree. We also show that, under the same
hypotheses, the number of unlabelled, -vertex graphs that are -locally
grows like a stretched exponential in , again in contrast with
-regular graphs. In the case where is the standard Cayley graph of
, we obtain a much more precise enumeration result, and more
precise results on the properties of the random graph . Our proofs
use a mixture of results and techniques from geometry, group theory and
combinatorics.Comment: Minor changes. 57 page
On the hyperbolicity of random graphs
Let be a connected graph with the usual (graph) distance metric
. Introduced by Gromov, is
-hyperbolic if for every four vertices , the two largest
values of the three sums differ
by at most . In this paper, we determinate the value of this
hyperbolicity for most binomial random graphs.Comment: 20 page
On giant components and treewidth in the layers model
Given an undirected -vertex graph and an integer , let
denote the random vertex induced subgraph of generated by ordering
according to a random permutation and including in those
vertices with at most of their neighbors preceding them in this order.
The distribution of subgraphs sampled in this manner is called the \emph{layers
model with parameter} . The layers model has found applications in studying
-degenerate subgraphs, the design of algorithms for the maximum
independent set problem, and in bootstrap percolation.
In the current work we expand the study of structural properties of the
layers model.
We prove that there are -regular graphs for which with high
probability has a connected component of size . Moreover,
this connected component has treewidth . This lower bound on the
treewidth extends to many other random graph models. In contrast, is
known to be a forest (hence of treewidth~1), and we establish that if is of
bounded degree then with high probability the largest connected component in
is of size . We also consider the infinite two-dimensional
grid, for which we prove that the first four layers contain a unique infinite
connected component with probability
Extremal Properties of Complex Networks
We describe the structure of connected graphs with the minimum and maximum
average distance, radius, diameter, betweenness centrality, efficiency and
resistance distance, given their order and size. We find tight bounds on these
graph qualities for any arbitrary number of nodes and edges and analytically
derive the form and properties of such networks
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur
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