7 research outputs found
Random Regular Graphs are not Asymptotically Gromov Hyperbolic
In this paper we prove that random --regular graphs with have
traffic congestion of the order where is the number
of nodes and geodesic routing is used. We also show that these graphs are not
asymptotically --hyperbolic for any non--negative almost
surely as .Comment: 6 pages, 2 figure
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
Geodesics and Almost Geodesic Cycles in Random Regular Graphs
A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d(G)(u, v) is at least d(C)(u, v) - e(n). Let omega(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= log(d-1)log(d-1) n+omega(n) and vertical bar C vertical bar =2 log(d-1) n+O(omega(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 66: 115-136, 2011FAPESP[2007/56496-3]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)MITACSMITACSNSERCNSERCCanadian Research Chairs ProgramCanadian Research Chairs Progra
Properties of graphs with large girth
This thesis is devoted to the analysis of a class of
iterative probabilistic algorithms in regular graphs, called
locally greedy algorithms, which will provide bounds for
graph functions in regular graphs with large girth. This class is
useful because, by conveniently setting the parameters associated
with it, we may derive algorithms for some well-known graph
problems, such as algorithms to find a large independent set, a
large induced forest, or even a small dominating set in an input
graph G. The name ``locally greedy" comes from the fact that, in
an algorithm of this class, the probability associated with the
random selection of a vertex v is determined by the current
state of the vertices within some fixed distance of v.
Given r > 2 and an r-regular graph G, we determine the
expected performance of a locally greedy algorithm in G,
depending on the girth g of the input and on the degree r of
its vertices. When the girth of the graph is sufficiently large,
this analysis leads to new lower bounds on the independence number
of G and on the maximum number of vertices in an induced forest
in G, which, in both cases, improve the bounds previously known.
It also implies bounds on the same functions in graphs with large
girth and maximum degree r and in random regular graphs. As a
matter of fact, the asymptotic lower bounds on the cardinality of
a maximum induced forest in a random regular graph improve earlier
bounds, while, for independent sets, our bounds coincide with
asymptotic lower bounds first obtained by Wormald. Our result
provides an alternative proof of these bounds which avoids sharp
concentration arguments.
The main contribution of this work lies in the method presented
rather than in these particular new bounds. This method allows us,
in some sense, to directly analyse prioritised algorithms in
regular graphs, so that the class of locally greedy algorithms, or
slight modifications thereof, may be applied to a wider range of
problems in regular graphs with large girth