283 research outputs found
Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions
We calculate the partition function of the -state Potts model
exactly for strips of the square and triangular lattices of various widths
and arbitrarily great lengths , with a variety of boundary
conditions, and with and restricted to satisfy conditions corresponding
to the ferromagnetic phase transition on the associated two-dimensional
lattices. From these calculations, in the limit , we determine
the continuous accumulation loci of the partition function zeros in
the and planes. Strips of the honeycomb lattice are also considered. We
discuss some general features of these loci.Comment: 12 pages, 12 figure
Unicyclic Components in Random Graphs
The distribution of unicyclic components in a random graph is obtained
analytically. The number of unicyclic components of a given size approaches a
self-similar form in the vicinity of the gelation transition. At the gelation
point, this distribution decays algebraically, U_k ~ 1/(4k) for k>>1. As a
result, the total number of unicyclic components grows logarithmically with the
system size.Comment: 4 pages, 2 figure
Spanning Trees on Graphs and Lattices in d Dimensions
The problem of enumerating spanning trees on graphs and lattices is
considered. We obtain bounds on the number of spanning trees and
establish inequalities relating the numbers of spanning trees of different
graphs or lattices. A general formulation is presented for the enumeration of
spanning trees on lattices in dimensions, and is applied to the
hypercubic, body-centered cubic, face-centered cubic, and specific planar
lattices including the kagom\'e, diced, 4-8-8 (bathroom-tile), Union Jack, and
3-12-12 lattices. This leads to closed-form expressions for for these
lattices of finite sizes. We prove a theorem concerning the classes of graphs
and lattices with the property that
as the number of vertices , where is a finite
nonzero constant. This includes the bulk limit of lattices in any spatial
dimension, and also sections of lattices whose lengths in some dimensions go to
infinity while others are finite. We evaluate exactly for the
lattices we considered, and discuss the dependence of on d and the
lattice coordination number. We also establish a relation connecting to the free energy of the critical Ising model for planar lattices .Comment: 28 pages, latex, 1 postscript figure, J. Phys. A, in pres
Dynamics of Triangulations
We study a few problems related to Markov processes of flipping
triangulations of the sphere. We show that these processes are ergodic and
mixing, but find a natural example which does not satisfy detailed balance. In
this example, the expected distribution of the degrees of the nodes seems to
follow the power law
-Stars or On Extending a Drawing of a Connected Subgraph
We consider the problem of extending the drawing of a subgraph of a given
plane graph to a drawing of the entire graph using straight-line and polyline
edges. We define the notion of star complexity of a polygon and show that a
drawing of an induced connected subgraph can be extended with at
most bends per edge, where is the
largest star complexity of a face of and is the size of the
largest face of . This result significantly improves the previously known
upper bound of [5] for the case where is connected. We also show
that our bound is worst case optimal up to a small additive constant.
Additionally, we provide an indication of complexity of the problem of testing
whether a star-shaped inner face can be extended to a straight-line drawing of
the graph; this is in contrast to the fact that the same problem is solvable in
linear time for the case of star-shaped outer face [9] and convex inner face
[13].Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
On Binary Matroid Minors and Applications to Data Storage over Small Fields
Locally repairable codes for distributed storage systems have gained a lot of
interest recently, and various constructions can be found in the literature.
However, most of the constructions result in either large field sizes and hence
too high computational complexity for practical implementation, or in low rates
translating into waste of the available storage space. In this paper we address
this issue by developing theory towards code existence and design over a given
field. This is done via exploiting recently established connections between
linear locally repairable codes and matroids, and using matroid-theoretic
characterisations of linearity over small fields. In particular, nonexistence
can be shown by finding certain forbidden uniform minors within the lattice of
cyclic flats. It is shown that the lattice of cyclic flats of binary matroids
have additional structure that significantly restricts the possible locality
properties of -linear storage codes. Moreover, a collection of
criteria for detecting uniform minors from the lattice of cyclic flats of a
given matroid is given, which is interesting in its own right.Comment: 14 pages, 2 figure
Inductive Construction of 2-Connected Graphs for Calculating the Virial Coefficients
In this paper we give a method for constructing systematically all simple
2-connected graphs with n vertices from the set of simple 2-connected graphs
with n-1 vertices, by means of two operations: subdivision of an edge and
addition of a vertex. The motivation of our study comes from the theory of
non-ideal gases and, more specifically, from the virial equation of state. It
is a known result of Statistical Mechanics that the coefficients in the virial
equation of state are sums over labelled 2-connected graphs. These graphs
correspond to clusters of particles. Thus, theoretically, the virial
coefficients of any order can be calculated by means of 2-connected graphs used
in the virial coefficient of the previous order. Our main result gives a method
for constructing inductively all simple 2-connected graphs, by induction on the
number of vertices. Moreover, the two operations we are using maintain the
correspondence between graphs and clusters of particles.Comment: 23 pages, 5 figures, 3 table
Families of Graphs With Chromatic Zeros Lying on Circles
We define an infinite set of families of graphs, which we call -wheels and
denote , that generalize the wheel () and biwheel ()
graphs. The chromatic polynomial for is calculated, and
remarkably simple properties of the chromatic zeros are found: (i) the real
zeros occur at for even and for odd;
and (ii) the complex zeros all lie, equally spaced, on the unit circle
in the complex plane. In the limit, the zeros
on this circle merge to form a boundary curve separating two regions where the
limiting function is analytic, viz., the exterior and
interior of the above circle. Connections with statistical mechanics are noted.Comment: 8 pages, Late
Ergodicity and Slowing Down in Glass-Forming Systems with Soft Potentials: No Finite-Temperature Singularities
The aim of this paper is to discuss some basic notions regarding generic
glass forming systems composed of particles interacting via soft potentials.
Excluding explicitly hard-core interaction we discuss the so called `glass
transition' in which super-cooled amorphous state is formed, accompanied with a
spectacular slowing down of relaxation to equilibrium, when the temperature is
changed over a relatively small interval. Using the classical example of a
50-50 binary liquid of N particles with different interaction length-scales we
show that (i) the system remains ergodic at all temperatures. (ii) the number
of topologically distinct configurations can be computed, is temperature
independent, and is exponential in N. (iii) Any two configurations in phase
space can be connected using elementary moves whose number is polynomially
bounded in N, showing that the graph of configurations has the `small world'
property. (iv) The entropy of the system can be estimated at any temperature
(or energy), and there is no Kauzmann crisis at any positive temperature. (v)
The mechanism for the super-Arrhenius temperature dependence of the relaxation
time is explained, connecting it to an entropic squeeze at the glass
transition. (vi) There is no Vogel-Fulcher crisis at any finite temperature T>0Comment: 10 pages, 9 figures, submitted to PR
Some Exact Results on the Potts Model Partition Function in a Magnetic Field
We consider the Potts model in a magnetic field on an arbitrary graph .
Using a formula of F. Y. Wu for the partition function of this model as a
sum over spanning subgraphs of , we prove some properties of concerning
factorization, monotonicity, and zeros. A generalization of the Tutte
polynomial is presented that corresponds to this partition function. In this
context we formulate and discuss two weighted graph-coloring problems. We also
give a general structural result for for cyclic strip graphs.Comment: 5 pages, late
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