496 research outputs found
A Lexicographic Product Cancellation Property for Digraphs
There are four prominent product graphs in graph theory: Cartesian, strong, direct, and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted , for some digraphs and . The paper concludes by proving a cancellation property for the lexicographic product of digraphs , , , and : if and , then . It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments
Cancellation Properties of Direct Products of Digraphs
This paper discusses the direct product cancellation of digraphs. We define the exact conditions on G such that GxK=HxK implies G=H. We focus first on simple equations such as GxK_2=HxK_2 where K_2 denotes a single arc and then extend this to the more general situation, GxK = HxK. Our results are achieved by using a âfactorialâ operation on graphs, which is in some sense analogous to the factorial of an integer
On cohomology theory of (di)graphs
To a digraph with a choice of certain integral basis, we construct a CW
complex, whose integral singular cohomology is canonically isomorphic to the
path cohomology of the digraph as introduced in \cite{GLMY}. The homotopy type
of the CW complex turns out to be independent of the choice of basis. After a
very brief discussion of functoriality, this construction immediately implies
some of the expected but perhaps combinatorially subtle properties of the
digraph cohomology and homotopy proved very recently \cite{GLMY2}. Furthermore,
one gets a very simple expected formula for the cup product of forms on the
digraph. On the other hand, we present an approach of using sheaf theory to
reformulate (di)graph cohomologies. The investigation of the path cohomology
from this framework, leads to a subtle version of Poincare lemma for digraphs,
which follows from the construction of the CW complex.Comment: 17 page
Positivity for Gaussian graphical models
Gaussian graphical models are parametric statistical models for jointly
normal random variables whose dependence structure is determined by a graph. In
previous work, we introduced trek separation, which gives a necessary and
sufficient condition in terms of the graph for when a subdeterminant is zero
for all covariance matrices that belong to the Gaussian graphical model. Here
we extend this result to give explicit cancellation-free formulas for the
expansions of nonzero subdeterminants.Comment: 16 pages, 3 figure
Decomposition and factorisation of transients in Functional Graphs
Functional graphs (FGs) model the graph structures used to analyze the
behavior of functions from a discrete set to itself. In turn, such functions
are used to study real complex phenomena evolving in time. As the systems
involved can be quite large, it is interesting to decompose and factorize them
into several subgraphs acting together. Polynomial equations over functional
graphs provide a formal way to represent this decomposition and factorization
mechanism, and solving them validates or invalidates hypotheses on their
decomposability. The current solution method breaks down a single equation into
a series of \emph{basic} equations of the form (with , ,
and being FGs) to identify the possible solutions. However, it is able to
consider just FGs made of cycles only. This work proposes an algorithm for
solving these basic equations for general connected FGs. By exploiting a
connection with the cancellation problem, we prove that the upper bound to the
number of solutions is closely related to the size of the cycle in the
coefficient of the equation. The cancellation problem is also involved in
the main algorithms provided by the paper. We introduce a polynomial-time
semi-decision algorithm able to provide constraints that a potential solution
will have to satisfy if it exists. Then, exploiting the ideas introduced in the
first algorithm, we introduce a second exponential-time algorithm capable of
finding all solutions by integrating several `hacks' that try to keep the
exponential as tight as possible
Amenability and geometry of semigroups
We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup
- âŚ