688 research outputs found

    Enumerating Colorings, Tensions and Flows in Cell Complexes

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    We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex XX, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in Z/kZ\mathbb{Z}/k\mathbb{Z} for some kk) or integral (with values in {−k+1,
,k−1}\{-k+1,\dots,k-1\}). 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 XX.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory Series

    One vertex spin-foams with the Dipole Cosmology boundary

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    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

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    A projekt cĂ©lkitƱzĂ©seit sikerĂŒlt megvalĂłsĂ­tani. A nĂ©gy Ă©v sorĂĄn több mint szĂĄz kivĂĄlĂł eredmĂ©ny szĂŒletett, amibƑl 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©rƑ, hogy a rĂ©sztvevƑk 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

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    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 pp-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which pp the graph is pp-colorable, and that a pp-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group Γ(p,m,k)\Gamma(p,m,k). 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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