165 research outputs found
Degree/diameter problem for mixed graphs
The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs
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Diameter, Girth And Other Properties Of Highly Symmetric Graphs
We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry.
In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups.
In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases.
In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date.
We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups.
Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties
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Extremal Metric and Topological Properties of Vertex Transitive and Cayley Graphs
We shall consider problems in two broad areas of mathematics, namely the area of the degree diameter problem and the area of regular maps.
In the degree diameter problem we investigate finding graphs as large as possible with a given degree and diameter. Further, we may consider additional properties of such extremal graphs, for example restrictions on the kinds of symmetries that the graph in question exhibits.
We provide two pieces of research relating to the degree diameter problem. First, we provide a new derivation of the Hoffman-Singleton graph and show that this derivation may be used with minor modification to derive the Bosák graph. Ultimately we show that no further natural modification of the construction we use can derive any other Moore or mixed-Moore graphs. Second, we answer the previously open question of whether the Gómez graphs, which are known to be vertex-transitive, are in addition also Cayley. In doing this, we also generalise the construction of the Gómez graphs and show that the Gómez graphs are the largest graphs for given degree and diameter following the generalised construction.
We also provide two pieces of research relating to regular maps. We aim to address the related questions of for which triples of parameters k, l and m there exist finite regular maps of face length k, vertex order l and Petrie walk length m. We then address the related question of determining for which n there exist regular maps which are self dual and self Petrie dual which have face length, vertex order and Petrie dual walk length n. We address both questions by constructions of regular maps in fractional linear groups, necessarily leading us to study some interesting related number theoretic questions
Quantum speedup of classical mixing processes
Most approximation algorithms for #P-complete problems (e.g., evaluating the
permanent of a matrix or the volume of a polytope) work by reduction to the
problem of approximate sampling from a distribution over a large set
. This problem is solved using the {\em Markov chain Monte Carlo} method: a
sparse, reversible Markov chain on with stationary distribution
is run to near equilibrium. The running time of this random walk algorithm, the
so-called {\em mixing time} of , is as shown
by Aldous, where is the spectral gap of and is the minimum
value of . A natural question is whether a speedup of this classical
method to , the diameter of the graph
underlying , is possible using {\em quantum walks}.
We provide evidence for this possibility using quantum walks that {\em
decohere} under repeated randomized measurements. We show: (a) decoherent
quantum walks always mix, just like their classical counterparts, (b) the
mixing time is a robust quantity, essentially invariant under any smooth form
of decoherence, and (c) the mixing time of the decoherent quantum walk on a
periodic lattice is , which is indeed
and is asymptotically no worse than the
diameter of (the obvious lower bound) up to at most a logarithmic
factor.Comment: 13 pages; v2 revised several part
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Extremal Directed And Mixed Graphs
We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs.
A directed graph or mixed graph is -geodetic if there is no pair of vertices of such that there exist distinct non-backtracking walks with length in from to . The order of a -geodetic digraph with minimum out-degree is bounded below by the \emph{directed Moore bound} ; similarly the order of a -geodetic mixed graph with minimum undirected degree and minimum directed out-degree is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} .
The \emph{degree/geodecity problem} asks for the smallest possible order of a -geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure.
We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages.
We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular.
Using counting arguments we then provide strong bounds on the order of totally regular -geodetic mixed graphs and use these results to derive new extremal mixed graphs.
Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a -geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected -geodetic digraphs, as well as providing constructions of strongly connected -geodetic digraphs that we conjecture to be extremal for larger . We close with a discussion of some related generalised Tur\'{a}n problems for -geodetic digraphs
Glassy dynamics of kinetically constrained models
We review the use of kinetically constrained models (KCMs) for the study of
dynamics in glassy systems. The characteristic feature of KCMs is that they
have trivial, often non-interacting, equilibrium behaviour but interesting slow
dynamics due to restrictions on the allowed transitions between configurations.
The basic question which KCMs ask is therefore how much glassy physics can be
understood without an underlying ``equilibrium glass transition''. After a
brief review of glassy phenomenology, we describe the main model classes, which
include spin-facilitated (Ising) models, constrained lattice gases, models
inspired by cellular structures such as soap froths, models obtained via
mappings from interacting systems without constraints, and finally related
models such as urn, oscillator, tiling and needle models. We then describe the
broad range of techniques that have been applied to KCMs, including exact
solutions, adiabatic approximations, projection and mode-coupling techniques,
diagrammatic approaches and mappings to quantum systems or effective models.
Finally, we give a survey of the known results for the dynamics of KCMs both in
and out of equilibrium, including topics such as relaxation time divergences
and dynamical transitions, nonlinear relaxation, aging and effective
temperatures, cooperativity and dynamical heterogeneities, and finally
non-equilibrium stationary states generated by external driving. We conclude
with a discussion of open questions and possibilities for future work.Comment: 137 pages. Additions to section on dynamical heterogeneities (5.5,
new pages 110 and 112), otherwise minor corrections, additions and reference
updates. Version to be published in Advances in Physic
Synchronization in complex networks
Synchronization processes in populations of locally interacting elements are
in the focus of intense research in physical, biological, chemical,
technological and social systems. The many efforts devoted to understand
synchronization phenomena in natural systems take now advantage of the recent
theory of complex networks. In this review, we report the advances in the
comprehension of synchronization phenomena when oscillating elements are
constrained to interact in a complex network topology. We also overview the new
emergent features coming out from the interplay between the structure and the
function of the underlying pattern of connections. Extensive numerical work as
well as analytical approaches to the problem are presented. Finally, we review
several applications of synchronization in complex networks to different
disciplines: biological systems and neuroscience, engineering and computer
science, and economy and social sciences.Comment: Final version published in Physics Reports. More information
available at http://synchronets.googlepages.com
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