Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Representation of finite graphs as difference graphs of S-units, I   Let G be a simple finite graph such that each vertex has an integer value and
different vertices have different values. Let S be a finite non-empty set of
primes. We call G an S-graph if any two vertices are connected by an edge if
and only their values differ by a number which is composed of primes from S. We
prove e.g. that for every G there exist infinitely many finite sets S such that
G is an S-graph. We deal with cycles and complete bipartite graphs G. We
consider the triangles in G for a deeper analysis. Finally we prove that G is
an S-graph for all S if and only if G is cubical. Besides combinatorial and
numbertheoretical arguments some deep Diophantine results concerning S-unit
equations are used in our proofs.

29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th