1,284 research outputs found
Sumset Valuations of Graphs and Their Applications
International audienc
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs
A gain graph is a graph whose edges are orientably labelled from a group. A
weighted gain graph is a gain graph with vertex weights from an abelian
semigroup, where the gain group is lattice ordered and acts on the weight
semigroup. For weighted gain graphs we establish basic properties and we
present general dichromatic and forest-expansion polynomials that are Tutte
invariants (they satisfy Tutte's deletion-contraction and multiplicative
identities). Our dichromatic polynomial includes the classical graph one by
Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with
positive integer weights, and that of rooted integral gain graphs by Forge and
Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that
remains to be found.
An evaluation of one example of our polynomial counts proper list colorations
of the gain graph from a color set with a gain-group action. When the gain
group is Z^d, the lists are order ideals in the integer lattice Z^d, and there
are specified upper bounds on the colors, then there is a formula for the
number of bounded proper colorations that is a piecewise polynomial function of
the upper bounds, of degree nd where n is the order of the graph.
This example leads to graph-theoretical formulas for the number of integer
lattice points in an orthotope but outside a finite number of affinographic
hyperplanes, and for the number of n x d integral matrices that lie between two
specified matrices but not in any of certain subspaces defined by simple row
equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added
references, clarified examples. 35 p
On the asymptotics of dimers on tori
We study asymptotics of the dimer model on large toric graphs. Let be a weighted -periodic planar graph, and let
be a large-index sublattice of . For bipartite we
show that the dimer partition function on the quotient
has the asymptotic expansion , where is the area of ,
is the free energy density in the bulk, and is a finite-size
correction term depending only on the conformal shape of the domain together
with some parity-type information. Assuming a conjectural condition on the zero
locus of the dimer characteristic polynomial, we show that an analogous
expansion holds for non-bipartite. The functional form of the
finite-size correction differs between the two classes, but is universal within
each class. Our calculations yield new information concerning the distribution
of the number of loops winding around the torus in the associated double-dimer
models.Comment: 48 pages, 18 figure
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Potentials Unbounded Below
Continuous interpolates are described for classical dynamical systems defined
by discrete time-steps. Functional conjugation methods play a central role in
obtaining the interpolations. The interpolates correspond to particle motion in
an underlying potential, . Typically, has no lower bound and can exhibit
switchbacks wherein changes form when turning points are encountered by the
particle. The Beverton-Holt and Skellam models of population dynamics, and
particular cases of the logistic map are used to illustrate these features.Comment: Based on a talk given 29 July 2010, at the workshop on Supersymmetric
Quantum Mechanics and Spectral Design, Centro de Ciencias de Benasque Pedro
Pascual. This version incorporates modifications to conform to the published
paper: Additional references and discussion; New section 3.2 on the Skellam
exponential model; Appendix change
Spectral computations on lamplighter groups and Diestel-Leader graphs
The Diestel-Leader graph DL(q,r) is the horocyclic product of the homogeneous
trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of
the lamplighter group (wreath product of the cyclic group of order q with the
infinite cyclic group) with respect to a natural generating set. For the
"Simple random walk" (SRW) operator on the latter group, Grigorchuk & Zuk and
Dicks & Schick have determined the spectrum and the (on-diagonal) spectral
measure (Plancherel measure). Here, we show that thanks to the geometric
realization, these results can be obtained for all DL-graphs by directly
computing an l^2-complete orthonormal system of finitely supported
eigenfunctions of the SRW. This allows computation of all matrix elements of
the spectral resolution, including the Plancherel measure. As one application,
we determine the sharp asymptotic behaviour of the N-step return probabilities
of SRW. The spectral computations involve a natural approximating sequence of
finite subgraphs, and we study the question whether the cumulative spectral
distributions of the latter converge weakly to the Plancherel measure. To this
end, we provide a general result regarding Foelner approximations; in the
specific case of DL(q,r), the answer is positive only when r=q
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