Summary: This paper deals with those positive integers $N$ such that, for given integers $g$ and $k$ with $2 \le k < g$, 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 $\lfloor $gcd(g - $k, k^{2} - 1)/(k + 1) \rfloor = m - 1$. 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].

11A63, 05C20

Young graph, reverse multiple, palintiple, radix representation