236 research outputs found

    Ternary numbers and algebras. Reflexive numbers and Berger graphs

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    The Calabi-Yau spaces with SU(m) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly related with the nn-ary algebras. To find such algebras we study the n-ary generalization of the well-known binary norm division algebras, R{\mathbb R}, C{\mathbb C}, H{\mathbb H}, O{\mathbb O}, which helped to discover the most important "minimal" binary simple Lie groups, U(1), SU(2) and G(2). As the most important example, we consider the case n=3n=3, which gives the ternary generalization of quaternions and octonions, 3p3^p, p=2,3p=2,3, respectively. The ternary generalization of quaternions is directly related to the new ternary algebra and group which are related to the natural extensions of the binary su(3)su(3) algebra and SU(3) group. Using this ternary algebra we found the solution for the Berger graph: a tetrahedron.Comment: Revised version with minor correction

    Inverse Graphs Associated with Finite Groups

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    Quantum walks: a comprehensive review

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    Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa

    Inverse Graphs Associated with Finite Groups

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    Let (Γ,)(\Gamma,*) be a finite group and SS a possibly empty subset of Γ\Gamma containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with Γ\Gamma whose set of vertices coincides with Γ\Gamma such that two distinct vertices uu and vv are adjacent if and only if either uvSu * v\in S or vuSv * u\in S. We then investigate its algebraic and combinatorial structures

    On prisms, M\"obius ladders and the cycle space of dense graphs

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    For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main purpose of this paper is to prove the following: for every s > 0 there exists n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all circuits of X having length either f_0(X)-1 or f_0(X) generates all of Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure

    Master index: volumes 31–40

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