236 research outputs found
Ternary numbers and algebras. Reflexive numbers and Berger graphs
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 -ary algebras. To find such algebras we study the
n-ary generalization of the well-known binary norm division algebras, , , , , 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 , which gives the ternary
generalization of quaternions and octonions, , , 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
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
Quantum walks: a comprehensive review
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
Let be a finite group and a possibly empty subset of containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with whose set of vertices coincides with such that two distinct vertices and are adjacent if and only if either or . We then investigate its algebraic and combinatorial structures
On prisms, M\"obius ladders and the cycle space of dense graphs
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
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