52,031 research outputs found
Cylindrical Graph Construction (definition and basic properties)
In this article we introduce the {\it cylindrical construction} for graphs
and investigate its basic properties. We state a main result claiming a weak
tensor-like duality for this construction. Details of our motivations and
applications of the construction will appear elsewhere
Threshold graph limits and random threshold graphs
We study the limit theory of large threshold graphs and apply this to a
variety of models for random threshold graphs. The results give a nice set of
examples for the emerging theory of graph limits.Comment: 47 pages, 8 figure
Skew Schubert polynomials
We define skew Schubert polynomials to be normal form (polynomial)
representatives of certain classes in the cohomology of a flag manifold. We
show that this definition extends a recent construction of Schubert polynomials
due to Bergeron and Sottile in terms of certain increasing labeled chains in
Bruhat order of the symmetric group. These skew Schubert polynomials expand in
the basis of Schubert polynomials with nonnegative integer coefficients that
are precisely the structure constants of the cohomology of the complex flag
variety with respect to its basis of Schubert classes. We rederive the
construction of Bergeron and Sottile in a purely combinatorial way, relating it
to the construction of Schubert polynomials in terms of rc-graphs.Comment: 10 pages, 7 figure
A note on the minimum skew rank of a graph
The minimum skew rank of a graph over a field
is the smallest possible rank among all skew symmetric matrices
over , whose (,)-entry (for ) is nonzero whenever
is an edge in and is zero otherwise. We give some new properties of
the minimum skew rank of a graph, including a characterization of the graphs
with cut vertices over the infinite field such that
, determination of the minimum skew rank of -paths
over a field , and an extending of an existing result to show that
for a connected graph
with no even cycles and a field , where is the matching
number of , and is the largest possible rank among
all skew symmetric matrices over
- β¦