2,780 research outputs found
On bipartite graphs of defect at most 4
We consider the bipartite version of the degree/diameter problem, namely,
given natural numbers {\Delta} \geq 2 and D \geq 2, find the maximum number
Nb({\Delta},D) of vertices in a bipartite graph of maximum degree {\Delta} and
diameter D. In this context, the Moore bipartite bound Mb({\Delta},D)
represents an upper bound for Nb({\Delta},D). Bipartite graphs of maximum
degree {\Delta}, diameter D and order Mb({\Delta},D), called Moore bipartite
graphs, have turned out to be very rare. Therefore, it is very interesting to
investigate bipartite graphs of maximum degree {\Delta} \geq 2, diameter D \geq
2 and order Mb({\Delta},D) - \epsilon with small \epsilon > 0, that is,
bipartite ({\Delta},D,-\epsilon)-graphs. The parameter \epsilon is called the
defect. This paper considers bipartite graphs of defect at most 4, and presents
all the known such graphs. Bipartite graphs of defect 2 have been studied in
the past; if {\Delta} \geq 3 and D \geq 3, they may only exist for D = 3.
However, when \epsilon > 2 bipartite ({\Delta},D,-\epsilon)-graphs represent a
wide unexplored area. The main results of the paper include several necessary
conditions for the existence of bipartite -graphs; the complete
catalogue of bipartite (3,D,-\epsilon)-graphs with D \geq 2 and 0 \leq \epsilon
\leq 4; the complete catalogue of bipartite ({\Delta},D,-\epsilon)-graphs with
{\Delta} \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq \epsilon \leq 4; and a
non-existence proof of all bipartite ({\Delta},D,-4)-graphs with {\Delta} \geq
3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs
of defect 4 for {\Delta} \geq 3 and D \geq 5, and comment on some implications
of our results for upper bounds of Nb({\Delta},D).Comment: 25 pages, 14 Postscript figure
On large bipartite graphs of diameter 3
We consider the bipartite version of the {\it degree/diameter problem},
namely, given natural numbers and , find the maximum number
of vertices in a bipartite graph of maximum degree and diameter
. In this context, the bipartite Moore bound \M^b(d,D) represents a
general upper bound for . Bipartite graphs of order \M^b(d,D) are
very rare, and determining still remains an open problem for most
pairs.
This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on
bipartite -graphs (that is, bipartite graphs of order \M^b(d,D)-4)
was carried out. Here we first present some structural properties of bipartite
-graphs, and later prove there are no bipartite -graphs.
This result implies that the known bipartite -graph is optimal, and
therefore . Our approach also bears a proof of the uniqueness of
the known bipartite -graph, and the non-existence of bipartite
-graphs.
In addition, we discover three new largest known bipartite (and also
vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which
improves by 4 vertices the previous lower bound for
Exact bosonization of the Ising model
We present exact combinatorial versions of bosonization identities, which
equate the product of two Ising correlators with a free field (bosonic)
correlator. The role of the discrete free field is played by the height
function of an associated bipartite dimer model. Some applications to the
asymptotic analysis of Ising correlators are discussed.Comment: 35 page
Height representation of XOR-Ising loops via bipartite dimers
The XOR-Ising model on a graph consists of random spin configurations on
vertices of the graph obtained by taking the product at each vertex of the
spins of two independent Ising models. In this paper, we explicitly relate loop
configurations of the XOR-Ising model and those of a dimer model living on a
decorated, bipartite version of the Ising graph. This result is proved for
graphs embedded in compact surfaces of genus g.
Using this fact, we then prove that XOR-Ising loops have the same law as
level lines of the height function of this bipartite dimer model. At
criticality, the height function is known to converge weakly in distribution to
a Gaussian free field.
As a consequence, results of this paper shed a light on the occurrence of the
Gaussian free field in the XOR-Ising model. In particular, they prove a
discrete analogue of Wilson's conjecture, stating that the scaling limit of
XOR-Ising loops are "contour lines" of the Gaussian free field.Comment: 41 pages, 10 figure
Towards Effective Exact Algorithms for the Maximum Balanced Biclique Problem
The Maximum Balanced Biclique Problem (MBBP) is a prominent model with
numerous applications. Yet, the problem is NP-hard and thus computationally
challenging. We propose novel ideas for designing effective exact algorithms
for MBBP. Firstly, we introduce an Upper Bound Propagation procedure to
pre-compute an upper bound involving each vertex. Then we extend an existing
branch-and-bound algorithm by integrating the pre-computed upper bounds. We
also present a set of new valid inequalities induced from the upper bounds to
tighten an existing mathematical formulation for MBBP. Lastly, we investigate
another exact algorithm scheme which enumerates a subset of balanced bicliques
based on our upper bounds. Experiments show that compared to existing
approaches, the proposed algorithms and formulations are more efficient in
solving a set of random graphs and large real-life instances
Worm Monte Carlo study of the honeycomb-lattice loop model
We present a Markov-chain Monte Carlo algorithm of "worm"type that correctly
simulates the O(n) loop model on any (finite and connected) bipartite cubic
graph, for any real n>0, and any edge weight, including the fully-packed limit
of infinite edge weight. Furthermore, we prove rigorously that the algorithm is
ergodic and has the correct stationary distribution. We emphasize that by using
known exact mappings when n=2, this algorithm can be used to simulate a number
of zero-temperature Potts antiferromagnets for which the Wang-Swendsen-Kotecky
cluster algorithm is non-ergodic, including the 3-state model on the
kagome-lattice and the 4-state model on the triangular-lattice. We then use
this worm algorithm to perform a systematic study of the honeycomb-lattice loop
model as a function of n<2, on the critical line and in the densely-packed and
fully-packed phases. By comparing our numerical results with Coulomb gas
theory, we identify the exact scaling exponents governing some fundamental
geometric and dynamic observables. In particular, we show that for all n<2, the
scaling of a certain return time in the worm dynamics is governed by the
magnetic dimension of the loop model, thus providing a concrete dynamical
interpretation of this exponent. The case n>2 is also considered, and we
confirm the existence of a phase transition in the 3-state Potts universality
class that was recently observed via numerical transfer matrix calculations.Comment: 33 pages, 12 figure
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