 
Summary: The Representation Number of Some Sparse Graphs
Reza Akhtar
Dept. of Mathematics
Miami University, Oxford, OH 45056, USA
reza@calico.mth.muohio.edu
September 5, 2011
Abstract
We study the representation number for specific families of sparse graphs; in
particular, we give an exact formula for graphs with a single edge and complete
binary trees and an improved lower bound for the representation number of the
hypercube. We also study the prime factorization of the representation number
of graphs with a single edge.
1 Introduction
A finite graph G is said to be representable modulo r if there exists an injective map
f : V (G) {0, 1, . . . , r  1} such that for all u, v V (G), gcd(f(u)  f(v), r) = 1 if
and only if uv E(G). This is equivalent to requiring that there exist an injective
map f : V (G) Zr such that for all u, v V (G), f(u)  f(v) is a unit of (the ring)
Zr if and only if uv E(G). The representation number of G, denoted rep(G), is the
smallest positive integer r modulo which G is representable. If we define the unitary
Cayley graph Cay(r) on r vertices to be the graph with vertex set {0, 1, . . . , r1}, two
