105 research outputs found
Enumeration of perfect matchings of a type of quadratic lattice on the torus
NSFC [10831001]A quadrilateral cylinder of length m and breadth n is the Cartesian product of a m-cycle(with m vertices) and a n-path(with n vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as x(1), x(2), ... , x(m) and y(1), y(2), ... , y(m), respectively, where x(i) corresponds to y(i) (i = 1, 2, ..., m). We denote by Q(m,n,r), the graph obtained from quadrilateral cylinder of length m and breadth n by adding edges x(i)y(i+r) (r is a integer, 0 <= r < m and i + r is modulo m). Kasteleyn had derived explicit expressions of the number of perfect matchings for Q(m,n,0) [P.W. Kasteleyn, The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961), 1209-1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressions of the number of perfect matchings for Q(m,n,r) by enumerating Pfaffians
Investigations in the semi-strong product of graphs and bootstrap percolation
The semi-strong product of graphs G and H is a way of forming a new graph from the graphs G and H. The vertex set of the semi-strong product is the Cartesian product of the vertex sets of G and H, V(G) x V(H). The edges of the semi-strong product are determined as follows: (g1,h1)(g2,h2) is an edge of the product whenever g1g2 is an edge of G and h1h2 is an edge of H or g1 = g2 and h1h2 is an edge of H.
A natural subject for investigation is to determine properties of the semi-strong product in terms of those properties of its factors. We investigate distance, independence, matching, and domination in the semi-strong product
Bootstrap Percolation is a process defined on a graph. We begin with an initial set of infected vertices. In each subsequent round, uninfected vertices become infected if they are adjacent to at least r infected vertices. Once infected, vertices remain infected. The parameter r is called the percolation threshold. When G is finite, the infection either stops at a proper subset of G or all of V(G) becomes infected. If all of V(G) eventually becomes infected, then we say that the infection percolates and we call the initial set of infected vertices a percolating set.
The cardinality of a minimum percolating set of G with percolation threshold r is denoted m(G,r). We determine m(G,r) for certain Kneser graphs and bipartite Kneser graphs
Dimer geometry, amoebae and a vortex dimer model
We present a geometrical approach for studying dimers. We introduce a
connection for dimer problems on bipartite and non-bipartite graphs. In the
bipartite case the connection is flat but has non-trivial holonomy
round certain curves. This holonomy has the universality property that it does
not change as the number of vertices in the fundamental domain of the graph is
increased. It is argued that the K-theory of the torus, with or without
punctures, is the appropriate underlying invariant. In the non-bipartite case
the connection has non-zero curvature as well as non-zero Chern number. The
curvature does not require the introduction of a magnetic field. The phase
diagram of these models is captured by what is known as an amoeba. We introduce
a dimer model with negative edge weights that give rise to vortices. The
amoebae for various models are studied with particular emphasis on the case of
negative edge weights which corresponds to the presence of vortices. Vortices
gives rise to new kinds of amoebae with certain singular structures which we
investigate. On the amoeba of the vortex full hexagonal lattice we find the
partition function corresponds to that of a massless Dirac doublet.Comment: 25 pages, 9 figures Latest version: some references added and typos
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Brane Tilings and Quiver Gauge Theories
This work presents recent developments on brane tilings and their vacuum moduli
spaces.
Brane tilings are bipartite periodic graphs on the torus and represent 4d N = 1
supersymmetric worldvolume theories living on D3-branes probing Calabi-Yau 3-fold
singularities. The graph and combinatorial properties of brane tilings make the set
of supersymmetric quiver theories represented by them one of the largest and richest
known so far. The aim of this work is to give a concise pedagogical introduction to brane
tilings and a summary on recent exciting advancement on their classification, dualities
and construction.
At first, particular focus is given on counting distinct Abelian orbifolds of the form
C3/[gamma]. The presented counting of Abelian orbifolds of C3 and in more general of CD gives a first insight on the rich combinatorial nature of brane tilings. Following the classification theme, the work proceeds with the identification of all brane tilings whose
mesonic moduli spaces as toric Calabi-Yau 3-folds are represented by reflexive polygons.
There are 16 of these special convex lattice polygons. It is shown that 30 brane tilings
are associated with them. Some of these brane tilings are related by a correspondence
known as toric duality.
The classification of brane tilings with reflexive toric diagrams led to the discovery
of a new correspondence between brane tilings which we call specular duality. The
new correspondence identifies brane tilings with the same master space - the combined
mesonic and baryonic moduli space. As a by-product, the new correspondence paves
the way for constructing brane tilings which are not confined to the torus but are on
Riemann surfaces with arbitrary genus. We give the first classification of genus 2 brane
tilings, illustrate the corresponding supersymmetric quiver theories and analyse their
vacuum moduli spaces.Open Acces
Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space
For many algorithmic problems on graphs of treewidth , a standard dynamic
programming approach gives an algorithm with time and space complexity
. It turns out that when one
considers the more restrictive parameter treedepth, it is often the case that a
variation of this technique can be used to reduce the space complexity to
polynomial, while retaining time complexity of the form
, where is the treedepth. This
transfer of methodology is, however, far from automatic. For instance, for
problems with connectivity constraints, standard dynamic programming techniques
give algorithms with time and space complexity on graphs of treewidth , but it is not clear how to
convert them into time-efficient polynomial space algorithms for graphs of low
treedepth.
Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a
certain class of problems with connectivity constraints can be solved in time
and space complexity . Recently,
Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the
Cut&Count technique can be also applied in the setting of treedepth, and it
gives algorithms with running time
and polynomial space usage. However, a number of important problems eluded such
a treatment, with the most prominent examples being Hamiltonian Cycle and
Longest Path.
In this paper we clarify the situation by showing that Hamiltonian Cycle,
Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit
-time and polynomial space algorithms on graphs of
treedepth . The algorithms are randomized Monte Carlo with only false
negatives.Comment: Presented at WG2020. 20 pages, 2 figure
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