Let $n>1$ integer, and let $\Gamma_n$ be a simple graph with verteces $\{0, 1, \dots, n-1\}$ and $a, b$ adjacent iff $\gcd(a-b,n)=1$ (or $\gcd(b-a,n)=1$ if $b>a$).
I got proof of the following statements:
If $n$ - prime, then $\Gamma_n$ is complete graph.
If $n=p^k$, then $\Gamma_n$ is complement of $pK_{p^{k-1}}$
If $n, m$ - relative prime integers, them $\Gamma_{mn} = \Gamma_n \times \Gamma_m$ (tensor product)
Are there other well-defined sequences of graphs with such multiplicative properties? Or is there a way to construct such a sequence based on a multiplicative function?