Young Graphs: 1089 et al

TL;DR

This paper extends the study of (g, k) reverse multiples by proving isomorphism conjectures for 1089 and complete graphs, advancing the understanding of their combinatorial and cyclic properties.

Contribution

It proves Sloane's isomorphism conjectures for specific classes of reverse multiple graphs and advances the theoretical framework of their combinatorial structures.

Findings

Proved isomorphism conjectures for 1089 graphs.

Confirmed isomorphism for complete graphs.

Enhanced understanding of cyclic graph structures.

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.