73,800 research outputs found

    On fractional realizations of graph degree sequences

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    We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope. We describe properties of the polytope vertices and characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.Comment: 18 pages, 4 figure

    On finite groups all of whose cubic Cayley graphs are integral

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    For any positive integer kk, let Gk\mathcal{G}_k denote the set of finite groups GG such that all Cayley graphs Cay(G,S){\rm Cay}(G,S) are integral whenever ∣S∣≤k|S|\le k. Esteˊ{\rm \acute{e}}lyi and Kovaˊ{\rm \acute{a}}cs \cite{EK14} classified Gk\mathcal{G}_k for each k≥4k\ge 4. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class G3\mathcal{G}_3 is characterized. As an application, the classification of Gk\mathcal{G}_k is obtained again, where k≥4k\ge 4.Comment: 11 pages, accepted by Journal of Algebra and its Applications on June 201

    Bi-Criteria and Approximation Algorithms for Restricted Matchings

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    In this work we study approximation algorithms for the \textit{Bounded Color Matching} problem (a.k.a. Restricted Matching problem) which is defined as follows: given a graph in which each edge ee has a color cec_e and a profit pe∈Q+p_e \in \mathbb{Q}^+, we want to compute a maximum (cardinality or profit) matching in which no more than wj∈Z+w_j \in \mathbb{Z}^+ edges of color cjc_j are present. This kind of problems, beside the theoretical interest on its own right, emerges in multi-fiber optical networking systems, where we interpret each unique wavelength that can travel through the fiber as a color class and we would like to establish communication between pairs of systems. We study approximation and bi-criteria algorithms for this problem which are based on linear programming techniques and, in particular, on polyhedral characterizations of the natural linear formulation of the problem. In our setting, we allow violations of the bounds wjw_j and we model our problem as a bi-criteria problem: we have two objectives to optimize namely (a) to maximize the profit (maximum matching) while (b) minimizing the violation of the color bounds. We prove how we can "beat" the integrality gap of the natural linear programming formulation of the problem by allowing only a slight violation of the color bounds. In particular, our main result is \textit{constant} approximation bounds for both criteria of the corresponding bi-criteria optimization problem

    The 1/N expansion of colored tensor models

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    In this paper we perform the 1/N expansion of the colored three dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with more and more complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S^3 contribute to the leading order in the large N limit.Comment: typos corrected, references update

    Integral Invariants of 3-Manifolds

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    This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer and those of Kontsevich.Comment: 39 pages, AMS-LaTeX, to appear in J. Diff. Geo

    Grassmann Integral Representation for Spanning Hyperforests

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    Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J. Phys.
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