1,466 research outputs found
Local algorithms, regular graphs of large girth, and random regular graphs
We introduce a general class of algorithms and supply a number of general
results useful for analysing these algorithms when applied to regular graphs of
large girth. As a result, we can transfer a number of results proved for random
regular graphs into (deterministic) results about all regular graphs with
sufficiently large girth. This is an uncommon direction of transfer of results,
which is usually from the deterministic setting to the random one. In
particular, this approach enables, for the first time, the achievement of
results equivalent to those obtained on random regular graphs by a powerful
class of algorithms which contain prioritised actions. As examples, we obtain
new upper or lower bounds on the size of maximum independent sets, minimum
dominating sets, maximum and minimum bisection, maximum -independent sets,
minimum -dominating sets and minimum connected and weakly-connected
dominating sets in -regular graphs with large girth.Comment: Third version: no changes were made to the file. We would like to
point out that this paper was split into two parts in the publication
process. General theorems are in a paper with the same title, accepted by
Combinatorica. The applications of Section 9 are in a paper entitled
"Properties of regular graphs with large girth via local algorithms",
published by JCTB, doi 10.1016/j.jctb.2016.07.00
New families of small regular graphs of girth 5
In this paper we are interested in the {\it{Cage Problem}} that consists in
constructing regular graphs of given girth and minimum order. We focus on
girth , where cages are known only for degrees . We construct
regular graphs of girth using techniques exposed by Funk [Note di
Matematica. 29 suppl.1, (2009) 91 - 114] and Abreu et al. [Discrete Math. 312
(2012), 2832 - 2842] to obtain the best upper bounds known hitherto. The tables
given in the introduction show the improvements obtained with our results.Comment: 19 pages, 2 figure
Extremal Infinite Graph Theory
We survey various aspects of infinite extremal graph theory and prove several
new results. The lead role play the parameters connectivity and degree. This
includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure
Invariant Gaussian processes and independent sets on regular graphs of large girth
We prove that every 3-regular, n-vertex simple graph with sufficiently large
girth contains an independent set of size at least 0.4361n. (The best known
bound is 0.4352n.) In fact, computer simulation suggests that the bound our
method provides is about 0.438n.
Our method uses invariant Gaussian processes on the d-regular tree that
satisfy the eigenvector equation at each vertex for a certain eigenvalue
\lambda. We show that such processes can be approximated by i.i.d. factors
provided that . We then use these approximations
for to produce factor of i.i.d. independent sets on
regular trees.Comment: 19 page
Lines on smooth polarized -surfaces
For each integer , we give a sharp bound on the number of lines
contained in a smooth complex -polarized -surface in
. In the two most interesting cases of sextics in
and octics in , the bounds are and ,
respectively, as conjectured in an earlier paper.Comment: Substantially revised; finer and more complete result
An Automatic Speedup Theorem for Distributed Problems
Recently, Brandt et al. [STOC'16] proved a lower bound for the distributed
Lov\'asz Local Lemma, which has been conjectured to be tight for sufficiently
relaxed LLL criteria by Chang and Pettie [FOCS'17]. At the heart of their
result lies a speedup technique that, for graphs of girth at least ,
transforms any -round algorithm for one specific LLL problem into a
-round algorithm for the same problem. We substantially improve on this
technique by showing that such a speedup exists for any locally checkable
problem , with the difference that the problem the inferred
-round algorithm solves is not (necessarily) the same problem as .
Our speedup is automatic in the sense that there is a fixed procedure that
transforms a description for into a description for and
reversible in the sense that any -round algorithm for can be
transformed into a -round algorithm for . In particular, for any
locally checkable problem with exact deterministic time complexity on graphs with nodes, maximum node degree , and
girth at least , there is a sequence of problems
with time complexities , that can be
inferred from .
As a first application of our generalized speedup, we solve a long-standing
open problem of Naor and Stockmeyer [STOC'93]: we show that weak -coloring
in odd-degree graphs cannot be solved in rounds, thereby
providing a matching lower bound to their upper bound
Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length
In this paper, we deal with time-invariant spatially coupled low-density
parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches
usually start from quasi-cyclic low-density parity-check (QC-LDPC) block codes
and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that
the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently,
the symbolic parity-check matrix, leads to codes with smaller syndrome former
constraint lengths with respect to the best solutions available in the
literature. We provide theoretical lower bounds on the syndrome former
constraint length for the most relevant families of SC-LDPC-CCs, under
constraints on the minimum length of cycles in their Tanner graphs. We also
propose new code design techniques that approach or achieve such theoretical
limits.Comment: 30 pages, 5 figures, accepted for publication in IEEE Transactions on
Communication
Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs
A finite simple graph is called a bi-Cayley graph over a group if it has
a semiregular automorphism group, isomorphic to which has two orbits on
the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups
have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014),
679--693). In this paper we consider the latter class of graphs and select
those in the class which are also arc-transitive. Furthermore, such a graph is
called -type when it is bipartite, and the bipartition classes are equal to
the two orbits of the respective semiregular automorphism group. A -type
graph can be represented as the graph where is a
subset of the vertex set of which consists of two copies of say
and and the edge set is . A
bi-Cayley graph is called a BCI-graph if for any bi-Cayley
graph
implies that for some and . It is also shown that every cubic connected arc-transitive
-type bi-Cayley graph over an abelian group is a BCI-graph
Approximation of the integrated density of states on sofic groups
In this paper we study spectral properties of self-adjoint operators on a
large class of geometries given via sofic groups. We prove that the associated
integrated densities of states can be approximated via finite volume analogues.
This is investigated in the deterministic as well as in the random setting. In
both cases we cover a wide range of operators including in particular unbounded
ones. The large generality of our setting allows to treat applications from
long-range percolation and the Anderson model. Our results apply to operators
on Z^d, amenable groups, residually finite groups and therefore in particular
to operators on trees. All convergence results are established without any
ergodic theorem at hand.Comment: 37 pages, 8 figure
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
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