Young Graphs: 1089 et al

Abstract

This paper deals with those positive integers N such that, for given integers g and k with 1< k<g, the base-g digits of N and kN appear in reverse order. Such N are called (g, k) reverse multiples. Anne Ludington Young, in 1992, developed a kind of tree reflecting properties of these numbers; N. J. A. Sloane, in 2013, modified these trees into directed graphs and introduced certain combinatoric methods to determine from these graphs the number of reverse multiples for given values of g and k with a given number of digits. We extend their work, proving Sloane's isomorphism conjectures for 1089 graphs and complete graphs, furthering his study of cyclic graphs, and proving a minor result on isomorphism.