Let the multiplication graph $n:m$ be the graph with points $0,\dots,m\text{ - } 1$ and a line between points $a$ and $b$ when $n\cdot a \equiv b\operatorname{mod} m$.
From the answers to two other questions I've learned why especially the multiplication graph $3:64$ has $4$-cycles, and that in general a multiplication graph $n:m$ has $k$-cycles iff $k = \operatorname{ord}_d(n)$ for some factor $d$ of $m$, with the multiplicative order $\operatorname{ord}_d(n)$ being the smallest $p$ with $n^p \equiv 1 \operatorname{mod}d$ for coprime $n,d$.
My question is:
Are there simpler necessary or sufficient conditions on $n$ and $m$ for $n:m$ having a $k$-cycle than by cumbersomely first finding the divisors of $m$ and then calculating for each of them $\operatorname{ord}_d(n)$?
This is, how the multiplication graphs containing $4$-cycles are distributed in the table of graphs for $n, m = 2,..,20$, $n < m$ (click the picture to enlarge):
Do you see a distribution pattern that could be exploited to get some simpler conditions?
Note that the smallest graphs with a $7$-cycle are $7:29$, $16:29$, $20:29$, $23:29$, $24:29$, $25:29$.
