Regular graphs with linearly many triangles

Pim van der Hoorn, Gabor Lippner, Elchanan Mossel

A $d$-regular graph on $n$ nodes has at most $T_{\max }=\frac{n}{3}\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular…