15,880 research outputs found
A Complete Characterization of all Magic Constants Arising from Distance Magic Graphs
A positive integer is called a magic constant if there is a graph
along with a bijective function from to first natural
numbers such that the weight of the vertex for
all . It is known that all odd positive integers greater equal and
the integer powers of , , are magic constants. In this
paper we characterise all positive integers which are magic constants
Permanents, Pfaffian orientations, and even directed circuits
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in
such a way that the permanent of A equals the determinant of the modified
matrix? When does a real square matrix have the property that every real matrix
with the same sign pattern (that is, the corresponding entries either have the
same sign or are both zero) is nonsingular? When is a hypergraph with n
vertices and n hyperedges minimally nonbipartite? When does a bipartite graph
have a "Pfaffian orientation"? Given a digraph, does it have no directed
circuit of even length? Given a digraph, does it have a subdivision with no
even directed circuit?
It is known that all of the above problems are equivalent. We prove a
structural characterization of the feasible instances, which implies a
polynomial-time algorithm to solve all of the above problems. The structural
characterization says, roughly speaking, that a bipartite graph has a Pfaffian
orientation if and only if it can be obtained by piecing together (in a
specified way) planar bipartite graphs and one sporadic nonplanar bipartite
graph.Comment: 47 pages, published versio
Rainbow eulerian multidigraphs and the product of cycles
An arc colored eulerian multidigraph with colors is rainbow eulerian if
there is an eulerian circuit in which a sequence of colors repeats. The
digraph product that refers the title was introduced by Figueroa-Centeno et al.
as follows: let be a digraph and let be a family of digraphs such
that for every . Consider any function
. Then the product is the
digraph with vertex set and if and only if and .
In this paper we use rainbow eulerian multidigraphs and permutations as a way
to characterize the -product of oriented cycles. We study the
behavior of the -product when applied to digraphs with unicyclic
components. The results obtained allow us to get edge-magic labelings of graphs
formed by the union of unicyclic components and with different magic sums.Comment: 12 pages, 5 figure
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Nonlocal Games and Quantum Permutation Groups
We present a strong connection between quantum information and quantum
permutation groups. Specifically, we define a notion of quantum isomorphisms of
graphs based on quantum automorphisms from the theory of quantum groups, and
then show that this is equivalent to the previously defined notion of quantum
isomorphism corresponding to perfect quantum strategies to the isomorphism
game. Moreover, we show that two connected graphs and are quantum
isomorphic if and only if there exists and that are
in the same orbit of the quantum automorphism group of the disjoint union of
and . This connection links quantum groups to the more concrete notion
of nonlocal games and physically observable quantum behaviours. We exploit this
link by using ideas and results from quantum information in order to prove new
results about quantum automorphism groups, and about quantum permutation groups
more generally. In particular, we show that asymptotically almost surely all
graphs have trivial quantum automorphism group. Furthermore, we use examples of
quantum isomorphic graphs from previous work to construct an infinite family of
graphs which are quantum vertex transitive but fail to be vertex transitive,
answering a question from the quantum group literature.
Our main tool for proving these results is the introduction of orbits and
orbitals (orbits on ordered pairs) of quantum permutation groups. We show that
the orbitals of a quantum permutation group form a coherent
configuration/algebra, a notion from the field of algebraic graph theory. We
then prove that the elements of this quantum orbital algebra are exactly the
matrices that commute with the magic unitary defining the quantum group. We
furthermore show that quantum isomorphic graphs admit an isomorphism of their
quantum orbital algebras which maps the adjacency matrix of one graph to that
of the other.Comment: 39 page
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