688 research outputs found
Enumerating Colorings, Tensions and Flows in Cell Complexes
We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex , generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in for
some ) or integral (with values in ). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series
One vertex spin-foams with the Dipole Cosmology boundary
We find all the spin-foams contributing in the first order of the vertex
expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole
Cosmology model. Our algorithm is general and provides spin-foams of
arbitrarily given, fixed: boundary and, respectively, a number of internal
vertices. We use the recently introduced Operator Spin-Network Diagrams
framework.Comment: 23 pages, 30 figure
HipergrĂĄfok = Hypergraphs
A projekt cĂ©lkitƱzĂ©seit sikerĂŒlt megvalĂłsĂtani. A nĂ©gy Ă©v sorĂĄn több mint szĂĄz kivĂĄlĂł eredmĂ©ny szĂŒletett, amibĆl eddig 84 dolgozat jelent meg a tĂ©ma legkivĂĄlĂłbb folyĂłirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. SzĂĄmos rĂ©gĂłta fennĂĄllĂł sejtĂ©st bebizonyĂtottunk, egĂ©sz rĂ©gi nyitott problĂ©mĂĄt megoldottunk hipergrĂĄfokkal kapcsolatban illetve kapcsolĂłdĂł terĂŒleteken. A problĂ©mĂĄk nĂ©melyike sok Ă©ve, olykor több Ă©vtizede nyitott volt. Nem egy közvetlen kutatĂĄsi eredmĂ©ny, de szintĂ©n bizonyos Ă©rtĂ©kmĂ©rĆ, hogy a rĂ©sztvevĆk egyike a NorvĂ©g KirĂĄlyi AkadĂ©mia tagja lett Ă©s elnyerte a Steele dĂjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize
Colorings, determinants and Alexander polynomials for spatial graphs
A {\em balanced} spatial graph has an integer weight on each edge, so that
the directed sum of the weights at each vertex is zero. We describe the
Alexander module and polynomial for balanced spatial graphs (originally due to
Kinoshita \cite{ki}), and examine their behavior under some common operations
on the graph. We use the Alexander module to define the determinant and
-colorings of a balanced spatial graph, and provide examples. We show that
the determinant of a spatial graph determines for which the graph is
-colorable, and that a -coloring of a graph corresponds to a
representation of the fundamental group of its complement into a metacyclic
group . We finish by proving some properties of the Alexander
polynomial.Comment: 14 pages, 7 figures; version 3 reorganizes the paper, shortens some
of the proofs, and improves the results related to representations in
metacyclic groups. This is the final version, accepted by Journal of Knot
Theory and its Ramification
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