424 research outputs found
-product and -threshold graphs
This paper is the continuation of the research of the author and his
colleagues of the {\it canonical} decomposition of graphs. The idea of the
canonical decomposition is to define the binary operation on the set of graphs
and to represent the graph under study as a product of prime elements with
respect to this operation. We consider the graph together with the arbitrary
partition of its vertex set into subsets (-partitioned graph). On the
set of -partitioned graphs distinguished up to isomorphism we consider the
binary algebraic operation (-product of graphs), determined by the
digraph . It is proved, that every operation defines the unique
factorization as a product of prime factors. We define -threshold graphs as
graphs, which could be represented as the product of one-vertex
factors, and the threshold-width of the graph as the minimum size of
such, that is -threshold. -threshold graphs generalize the classes of
threshold graphs and difference graphs and extend their properties. We show,
that the threshold-width is defined for all graphs, and give the
characterization of graphs with fixed threshold-width. We study in detail the
graphs with threshold-widths 1 and 2
The Hamilton-Waterloo Problem with even cycle lengths
The Hamilton-Waterloo Problem HWP asks for a
2-factorization of the complete graph or , the complete graph with
the edges of a 1-factor removed, into -factors and
-factors, where . In the case that and are both
even, the problem has been solved except possibly when
or when and are both odd, in which case necessarily . In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP for odd and whenever the obvious
necessary conditions hold, except possibly if ; and
; ; or . This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above
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
On functional module detection in metabolic networks
Functional modules of metabolic networks are essential for understanding the metabolism of an organism as a whole. With the vast amount of experimental data and the construction of complex and large-scale, often genome-wide, models, the computer-aided identification of functional modules becomes more and more important. Since steady states play a key role in biology, many methods have been developed in that context, for example, elementary flux modes, extreme pathways, transition invariants and place invariants. Metabolic networks can be studied also from the point of view of graph theory, and algorithms for graph decomposition have been applied for the identification of functional modules. A prominent and currently intensively discussed field of methods in graph theory addresses the Q-modularity. In this paper, we recall known concepts of module detection based on the steady-state assumption, focusing on transition-invariants (elementary modes) and their computation as minimal solutions of systems of Diophantine equations. We present the Fourier-Motzkin algorithm in detail. Afterwards, we introduce the Q-modularity as an example for a useful non-steady-state method and its application to metabolic networks. To illustrate and discuss the concepts of invariants and Q-modularity, we apply a part of the central carbon metabolism in potato tubers (Solanum tuberosum) as running example. The intention of the paper is to give a compact presentation of known steady-state concepts from a graph-theoretical viewpoint in the context of network decomposition and reduction and to introduce the application of Q-modularity to metabolic Petri net models
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