  We study the Cohen-Macaulay property of triangular graphs $T_n$. We show that
$T_2$, $T_3$ and $T_5$ are Cohen-Macaulay graphs, and that $T_4$, $T_6$, $T_8$
and $T_n$ are not Cohen-Macaulay graphs, for $n\geq 10$. Finally, we prove that
over fields of characteristic zero $T_7$ and $T_9$ are Cohen-Macaulay.
