13,368 research outputs found
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
-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
Module Extensions Over Classical Lie Superalgebras
We study certain filtrations of indecomposable injective modules over
classical Lie superalgebras, applying a general approach for noetherian rings
developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the
consequences of our analysis, suppose that is a complex classical simple
Lie superalgebra and that is an indecomposable injective -module with
nonzero (and so necessarily simple) socle . (Recall that every essential
extension of , and in particular every nonsplit extension of by a simple
module, can be formed from -subfactors of .) A direct transposition of
the Lie algebra theory to this setting is impossible. However, we are able to
present a finite upper bound, easily calculated and dependent only on , for
the number of isomorphism classes of simple highest weight -modules
appearing as -subfactors of .Comment: 20 page
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