A triangle-free graph is a graph containing no graph cycles of length three. A simple graph is triangle-free iff , where is the adjacency matrix
of the graph and
is the matrix trace.

The numbers of triangle-free simple graphs on , 2, 3, ... nodes are 1, 2, 3, 7, 14, 38, 107, 410, 1897,
... (OEIS A006785), the first few of which
are illustrated above.

The numbers of triangle-free simple connected graphs on , 2, 3, ... nodes are 1, 1, 1, 3, 6, 19, 59, 267, ... (OEIS
A024607), the first few of which are illustrated
above.