Non-existence of one-dimensional expanding graphs

Abstract

Expanding graphs are the basic building blocks used in constructions of graphs with special connectivity properties such as superconcentrators. The only known explicit method (Margulis[7], Gabber and Galil[5]) of constructing arbitrarily large expanding graphs with a linear number of edges, uses graphs whose edges are defined by a finite set of linear mappings restricted to a two-dimensional set, Zn × Zn, where Zn denotes the integers mod n. In this paper we prove that for any finite set of onedimensional linear mappings with rational coefficients, the graph they define by their restriction to Zn is not an expanding graph. We also show that shuffle exchange graphs can not be expanding graphs.