174,535 research outputs found
Limits of Ordered Graphs and their Applications
The emerging theory of graph limits exhibits an analytic perspective on
graphs, showing that many important concepts and tools in graph theory and its
applications can be described more naturally (and sometimes proved more easily)
in analytic language. We extend the theory of graph limits to the ordered
setting, presenting a limit object for dense vertex-ordered graphs, which we
call an \emph{orderon}. As a special case, this yields limit objects for
matrices whose rows and columns are ordered, and for dynamic graphs that expand
(via vertex insertions) over time. Along the way, we devise an ordered
locality-preserving variant of the cut distance between ordered graphs, showing
that two graphs are close with respect to this distance if and only if they are
similar in terms of their ordered subgraph frequencies. We show that the space
of orderons is compact with respect to this distance notion, which is key to a
successful analysis of combinatorial objects through their limits.
We derive several applications of the ordered limit theory in extremal
combinatorics, sampling, and property testing in ordered graphs. In particular,
we prove a new ordered analogue of the well-known result by Alon and Stav
[RS\&A'08] on the furthest graph from a hereditary property; this is the first
known result of this type in the ordered setting. Unlike the unordered regime,
here the random graph model with an ordering over the vertices is
\emph{not} always asymptotically the furthest from the property for some .
However, using our ordered limit theory, we show that random graphs generated
by a stochastic block model, where the blocks are consecutive in the vertex
ordering, are (approximately) the furthest. Additionally, we describe an
alternative analytic proof of the ordered graph removal lemma [Alon et al.,
FOCS'17].Comment: Added a new application: An Alon-Stav type result on the furthest
ordered graph from a hereditary property; Fixed and extended proof sketch of
the removal lemma applicatio
Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
Denoting as the chromatic polynomial for coloring an -vertex
graph with colors, and considering the limiting function , a fundamental question in graph theory is the
following: is analytic or not at the origin
of the plane? (where the complex generalization of is assumed). This
question is also relevant in statistical mechanics because
, where is the ground state entropy of the
-state Potts antiferromagnet on the lattice graph , and the
analyticity of at is necessary for the large- series
expansions of . Although is analytic at for many
, there are some for which it is not; for these, has no
large- series expansion. It is important to understand the reason for this
nonanalyticity. Here we give a general condition that determines whether or not
a particular is analytic at and explains the
nonanalyticity where it occurs. We also construct infinite families of graphs
with functions that are non-analytic at and investigate the
properties of these functions. Our results are consistent with the conjecture
that a sufficient condition for to be analytic at is
that is a regular lattice graph . (This is known not to be a
necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in
Phys. Rev.
Fastest mixing Markov chain on graphs with symmetries
We show how to exploit symmetries of a graph to efficiently compute the
fastest mixing Markov chain on the graph (i.e., find the transition
probabilities on the edges to minimize the second-largest eigenvalue modulus of
the transition probability matrix). Exploiting symmetry can lead to significant
reduction in both the number of variables and the size of matrices in the
corresponding semidefinite program, thus enable numerical solution of
large-scale instances that are otherwise computationally infeasible. We obtain
analytic or semi-analytic results for particular classes of graphs, such as
edge-transitive and distance-transitive graphs. We describe two general
approaches for symmetry exploitation, based on orbit theory and
block-diagonalization, respectively. We also establish the connection between
these two approaches.Comment: 39 pages, 15 figure
Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Postprint (updated version
Counting Integer flows in Networks
This paper discusses new analytic algorithms and software for the enumeration
of all integer flows inside a network. Concrete applications abound in graph
theory \cite{Jaeger}, representation theory \cite{kirillov}, and statistics
\cite{persi}. Our methods clearly surpass traditional exhaustive enumeration
and other algorithms and can even yield formulas when the input data contains
some parameters. These methods are based on the study of rational functions
with poles on arrangements of hyperplanes
Analytic evaluation of Feynman graph integrals
We review the main steps of the differential equation approach to the
analytic evaluation of Feynman graphs, showing at the same time its application
to the 3-loop sunrise graph in a particular kinematical configuration.Comment: 5 pages, 1 figure, uses npb.sty. Presented at RADCOR 2002 and Loops
and Legs in Quantum Field Theory, 8-13 September 2002, Kloster Banz, Germany.
Revised version: minor typos corrected, one reference adde
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