Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Graph decompositions in projective geometries   Let PG$(\mathbb{F}_q^v)$ be the $(v-1)$-dimensional projective space over
$\mathbb{F}_q$ and let $\Gamma$ be a simple graph of order ${q^k-1\over q-1}$
for some $k$. A 2$-(v,\Gamma,\lambda)$ design over $\mathbb{F}_q$ is a
collection $\cal B$ of graphs (\textit{blocks}) isomorphic to $\Gamma$ with the
following properties: the vertex set of every block is a subspace of
PG$(\mathbb{F}_q^v)$; every two distinct points of PG$(\mathbb{F}_q^v)$ are
adjacent in exactly $\lambda$ blocks. This new definition covers, in
particular, the well known concept of a 2$-(v,k,\lambda)$ design over
$\mathbb{F}_q$ corresponding to the case that $\Gamma$ is complete.
  In this work of a foundational nature we illustrate how difference methods
allow us to get concrete non-trivial examples of $\Gamma$-decompositions over
$\mathbb{F}_2$ or $\mathbb{F}_3$ for which $\Gamma$ is a cycle, a path, a
prism, a generalized Petersen graph, or a Moebius ladder. In particular, we
will discuss in detail the special and very hard case that $\Gamma$ is complete
and $\lambda=1$, i.e., the Steiner 2-designs over a finite field. Also, we
briefly touch the new topic of near resolvable 2-$(v,2,1)$ designs over
$\mathbb{F}_q$.
  This study has led us to some (probably new) collateral problems concerning
difference sets. Supported by multiple examples, we conjecture the existence of
infinite families of $\Gamma$-decompositions over a finite field that can be
obtained by suitably labeling the vertices of $\Gamma$ with the elements of a
Singer difference set.

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The categories of this paper are: eess
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
1
The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: econ