688 research outputs found
Prime Graphs and Exponential Composition of Species
In this paper, we enumerate prime graphs with respect to the Cartesian
multiplication of graphs. We use the unique factorization of a connected graph
into the product of prime graphs given by Sabidussi to find explicit formulas
for labeled and unlabeled prime graphs. In the case of species, we construct
the exponential composition of species based on the arithmetic product of
species of Maia and M\'endez and the quotient species, and express the species
of connected graphs as the exponential composition of the species of prime
graphs.Comment: 30 pages, 7 figures, 1 tabl
The number of distinguishing colorings of a Cartesian product graph
A vertex coloring is called distinguishing if the identity is the only
automorphism that can preserve it. The distinguishing threshold of
a graph is the minimum number of colors required that any arbitrary
-coloring of is distinguishing. In this paper, we calculate the
distinguishing threshold of a Cartesian product graph. Moreover, we calculate
the number of non-equivalent distinguishing colorings of grids.Comment: 11 pages, 4 figure
Direct Product Primality Testing of Graphs is GI-hard
We investigate the computational complexity of the graph primality testing
problem with respect to the direct product (also known as Kronecker, cardinal
or tensor product). In [1] Imrich proves that both primality testing and a
unique prime factorization can be determined in polynomial time for (finite)
connected and nonbipartite graphs. The author states as an open problem how
results on the direct product of nonbipartite, connected graphs extend to
bipartite connected graphs and to disconnected ones. In this paper we partially
answer this question by proving that the graph isomorphism problem is
polynomial-time many-one reducible to the graph compositeness testing problem
(the complement of the graph primality testing problem). As a consequence of
this result, we prove that the graph isomorphism problem is polynomial-time
Turing reducible to the primality testing problem. Our results show that
connectedness plays a crucial role in determining the computational complexity
of the graph primality testing problem
On the degree conjecture for separability of multipartite quantum states
We settle the so-called degree conjecture for the separability of
multipartite quantum states, which are normalized graph Laplacians, first given
by Braunstein {\it et al.} [Phys. Rev. A \textbf{73}, 012320 (2006)]. The
conjecture states that a multipartite quantum state is separable if and only if
the degree matrix of the graph associated with the state is equal to the degree
matrix of the partial transpose of this graph. We call this statement to be the
strong form of the conjecture. In its weak version, the conjecture requires
only the necessity, that is, if the state is separable, the corresponding
degree matrices match. We prove the strong form of the conjecture for {\it
pure} multipartite quantum states, using the modified tensor product of graphs
defined in [J. Phys. A: Math. Theor. \textbf{40}, 10251 (2007)], as both
necessary and sufficient condition for separability. Based on this proof, we
give a polynomial-time algorithm for completely factorizing any pure
multipartite quantum state. By polynomial-time algorithm we mean that the
execution time of this algorithm increases as a polynomial in where is
the number of parts of the quantum system. We give a counter-example to show
that the conjecture fails, in general, even in its weak form, for multipartite
mixed states. Finally, we prove this conjecture, in its weak form, for a class
of multipartite mixed states, giving only a necessary condition for
separability.Comment: 17 pages, 3 figures. Comments are welcom
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