59 research outputs found
A Note on the Manickam-Mikl\'os-Singhi Conjecture for Vector Spaces
Let be an -dimensional vector space over a finite field
. Define a real-valued weight function on the -dimensional
vector spaces of such that the sum of all weights is zero. Let the weight
of a subspace be the sum of the weights of the -dimensional subspaces
contained in . In 1988 Manickam and Singhi conjectured that if ,
then the number of -dimensional subspaces with nonnegative weight is at
least the number of -dimensional subspaces on a fixed -dimensional
subspace.
Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the
conjecture of Manickam and Singhi for . We modify the technique used
by Chowdhury et al. to prove the conjecture for if is large.
Furthermore, if equality holds and , then the set of
-dimensional subspaces with nonnegative weight is the set of all
-dimensional subspaces on a fixed -dimensional subspace.Comment: 15 pages; this version fixes typos and some minor mistakes, also some
proofs got a bit more explicit for an easier understandin
Some non-existence results for distance- ovoids in small generalized polygons
We give a computer-based proof for the non-existence of distance- ovoids
in the dual split Cayley hexagon .
Furthermore, we give upper bounds on partial distance- ovoids of
for .Comment: 10 page
New strongly regular graphs from finite geometries via switching
We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type U(n, 2), O(n, 3), O(n, 5), O+ (n, 3), and O- (n, 3) are not determined by its parameters for n >= 6. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs. (C) 2019 Elsevier Inc. All rights reserved
Switching for Small Strongly Regular Graphs
We provide an abundance of strongly regular graphs (SRGs) for certain
parameters with . For this we use Godsil-McKay
(GM) switching with a partition of type and Wang-Qiu-Hu (WQH) switching
with a partition of type . In most cases, we start with a highly
symmetric graph which belongs to a finite geometry. To our knowledge, most of
the obtained graphs are new.
For all graphs, we provide statistics about the size of the automorphism
group. We also find the recently discovered Kr\v{c}adinac partial geometry,
therefore finding a third method of constructing it.Comment: 15 page
Regular Intersecting Families
We call a family of sets intersecting, if any two sets in the family
intersect. In this paper we investigate intersecting families of
-element subsets of such that every element of
lies in the same (or approximately the same) number of members of
. In particular, we show that we can guarantee if and only if .Comment: 15 pages, accepted versio
New Strongly Regular Graphs from Finite Geometries via Switching
We show that the strongly regular graph on non-isotropic points of one type
of the polar spaces of type , , , , and
are not determined by its parameters for . We prove this
by using a variation of Godsil-McKay switching recently described by Wang, Qiu,
and Hu. This also results in a new, shorter proof of a previous result of the
first author which showed that the collinearity graph of a polar space is not
determined by its spectrum. The same switching gives a linear algebra
explanation for the construction of a large number of non-isomorphic designs.Comment: 13 pages, accepted in Linear Algebra and Its Application
The Independence Number of the Orthogonality Graph in Dimension
We determine the independence number of the orthogonality graph on
-dimensional hypercubes. This answers a question by Galliard from 2001
which is motivated by a problem in quantum information theory. Our method is a
modification of a rank argument due to Frankl who showed the analogous result
for -dimensional hypercubes, where is an odd prime.Comment: 3 pages, accepted by Combinatorica, fixed a minor typo spotted by
Peter Si
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