asymmetric graph

An identity graph, sometimes also known as an asymmetric graph or rigid graph, is a graph possessing a single graph automorphism. The numbers of connected identity graphs on n = 1, 2, ... nodes are 1, 0, 0, 0, 0, 8, 144, 3552, 131452, ... (OEIS A124059), with the the eight identity graphs of order six (all of which are connected) illustrated above. The numbers of identity graphs on n = 1, 2, ... nodes are given by 1, 0, 0, 0, 0, 8, 152, 3696, 135004, ... (OEIS A003400), with the eight 7-node disconnected identity graphs illustrated above. The following table summarizes some named identity graphs, illustrated above.

automorphism group | graph automorphism | rigid graph | symmetric graph