These correspond to pure quadratic polynomials over the integers mod 2. The number of such graphs (up to isomorphism) with n vertices is well-known -- it is sequence A000088 in the On-Line Encyclopedia of Integer Sequences.

Here I have lists of the graphs in the following format. A graph on n vertices is represented by the upper half of its adjacency matrix, specifically as n-1 strings separated by spaces, where the i'th string has length i and the j'th letter of the string denotes whether vertex i+1 has an edge to vertex j. The different graphs are listed in lexicographic order applied to these strings.

- The 11 graphs with four vertices
- The 34 graphs with five vertices
- The 156 graphs with six vertices
- The 1044 graphs with seven vertices
- The 12346 graphs with eight vertices
- A listing of how many graphs with up to eight vertices extend each of the four-vertex and five-vertex graphs

Much more extensive listings, in another format, are available from Brendan McKay of Australian National University. (Hat tip again to Don Knuth.)

I am also interested in pure quadratic polynomials over other rings such as the integers mod 3. These are represented by graphs where the edges, when present, can have one of two colors, "1" or "2". The number of these with n vertices is also in the On-Line Encyclopedia as sequence A004102 -- these "signed graphs" were first enumerated by Harary in the 1950's. (Thanks to Don Knuth for this reference.) Here are lists, in a format similar to the above, of:

- The 10 colored graphs with three vertices
- The 66 colored graphs with four vertices
- The 792 colored graphs with five vertices
- The 25506 colored graphs with six vertices

Last modified 31 May 2007