A076184 Graph code numbers of simple graphs in numerical order.

0, 1, 3, 7, 11, 12, 13, 15, 30, 31, 63, 75, 76, 77, 79, 86, 87, 94, 95, 116, 117, 119, 127, 222, 223, 235, 236, 237, 239, 254, 255, 507, 511, 1023, 1099, 1100, 1101, 1103, 1108, 1109, 1110, 1111, 1118, 1119, 1140, 1141, 1143, 1151, 1182, 1183, 1184, 1185, 1187

Offset: 1

Author

K. R. Parthasarathy (nuns(AT)vsnl.com), Nov 02 2002

Keywords

• nonn

Comments

Every simple graph has a symmetric adjacency matrix whose lower triangular part by rows represents a little-endian binary number of which the minimum value over all isomorphic graphs gives the graph code number. Adding isolated vertices will not change the graph code number.

Study of the patterns and gaps in the sequence appears to be quite interesting.

The number of terms that are less than 2^(n*(n-1)/2) is equal to A000088(n). - Vladimir Kulipanov, Oct 13 2015

The numbers of vertices and edges of the graph with code a(n) are A002024(A029837(a(n)+1)) + 1 and A000120(a(n)), respectively. - Pontus von Brömssen, Mar 15 2025

To obtain all unlabeled graphs with n vertices (including graphs with isolated vertices), take the graphs associated with the first A000088(n) terms and add isolated vertices so that each graph gets n vertices. - Pontus von Brömssen, Mar 19 2025

Examples

			a(5)=11 in binary (with 0's prepended to give a triangular number of digits) is 001011 so adjacency matrix [0,1,1,1; 1,0,0,0; 1,0,0,0; 1,0,0,0].
a(6)=12 in binary is 001100 so adjacency matrix [0,0,0,1; 0,0,1,0; 0,1,0,0; 1,0,0,0].
		

References

• F. Harary, Problems involving graphical numbers, in Colloq. Math. Soc. Janos Bolyai, 4 (1970) 625-635. Look at his 'mincode numbers'.

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..12346 (for graphs on up to 8 vertices; terms 1..156 from Vladimir Kulipanov)
• Pontus von Brömssen, Python function to convert a code to a list of edges.
• K. R. Parthasarathy, Graph code numbers, preprint, 2002.

Crossrefs

Cf. A000088, A000120, A002024, A029837.