Representation of finite graphs as difference graphs of S-units, I

Let S be a finite non-empty set of primes, Z S the ring of those rationals whose denominators are not divisible by primes outside S, and Z S ⁎ the multiplicative group of invertible elements (S-units) in Z S . For a non-empty subset A of Z S , denote by G S ( A ) the graph with vertex set A and with an edge between a and b if and only if a − b ∈ Z S ⁎ . This type of graphs has been studied by many people. present paper we deal with the representability of finite (simple) graphs G as G S ( A ) . If A ′ = u A + a for some u ∈ Z S ⁎ and a ∈ Z S , then A and A ′ are called S-equivalent, since G S ( A ) and G S ( A ′ ) are isomorphic. We say that a finite graph G is representable/infinitely representable with S if G is isomorphic to G S ( A ) for some A/for infinitely many non-S-equivalent A. ve among other things that for any finite graph G there exist infinitely many finite sets S of primes such that G can be represented with S. We deal with the infinite representability of finite graphs, in particular cycles and complete bipartite graphs. Further, we consider the triangles in G for a deeper analysis. Finally, we prove that G is representable with every S if and only if G is cubical. s combinatorial and numbertheoretical arguments, some deep Diophantine results concerning S-unit equations are used in our proofs. t II, we shall investigate these and similar problems over more general domains.