Young Graphs: 1089 et al.

摘要

This paper deals with those positive integers N such that, for given integers g and k with 2 ≤ k g, the base-g digits of kN appear in reverse order from those of N. Such N are called (g, k) reverse multiples. Young, in 1992, developed a kind of tree reflecting properties of these numbers; Sloane, in 2013, modified these trees into directed graphs and introduced certain combinatorial 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 prove Sloanes isomorphism conjectures for 1089 graphs and complete graphs, namely that the Young graph for g and k is a 1089 graph if and only if k+1 | g and is a complete Young graph on m nodes if and only if ⌊ gcd(g - k, k2 - 1)/(k + 1) ⌋ = m - 1. We also extend his study of cyclic Young graphs and prove a minor result on isomorphism and the nodes adjacent to the node [0, 0].