OFFSET
1,18
COMMENTS
A graph in which every node has r edges is called an r-regular graph. The triangle is symmetric because if an n-node graph is r-regular, than its complement is (n - 1 - r)-regular and two graphs are isomorphic if and only if their complements are isomorphic.
Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A295193. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 08 2020
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24, first 16 rows from Jason Kimberley)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Markus Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [Jason Kimberley, Nov 24 2009]
Markus Meringer, Fast Generation of Regular Graphs and Construction of Cages, Univ. Bayreuth, 1999
Markus Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Regular Graph.
EXAMPLE
T(8,3) = 6. Edge-lists for the 6 3-regular 8-node graphs:
Graph 1: 12, 13, 14, 23, 24, 34, 56, 57, 58, 67, 68, 78
Graph 2: 12, 13, 14, 24, 34, 26, 37, 56, 57, 58, 68, 78
Graph 3: 12, 13, 23, 14, 47, 25, 58, 36, 45, 67, 68, 78
Graph 4: 12, 13, 23, 14, 25, 36, 47, 48, 57, 58, 67, 68
Graph 5: 12, 13, 24, 34, 15, 26, 37, 48, 56, 57, 68, 78
Graph 6: 12, 23, 34, 45, 56, 67, 78, 18, 15, 26, 37, 48.
Triangle starts
1;
1, 1;
1, 0, 1;
1, 1, 1, 1;
1, 0, 1, 0, 1;
1, 1, 2, 2, 1, 1;
1, 0, 2, 0, 2, 0, 1;
1, 1, 3, 6, 6, 3, 1, 1;
1, 0, 4, 0, 16, 0, 4, 0, 1;
1, 1, 5, 21, 60, 60, 21, 5, 1, 1;
1, 0, 6, 0, 266, 0, 266, 0, 6, 0, 1;
1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1;
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
More terms and comments from David Wasserman, Feb 22 2002
More terms from Eric W. Weisstein, Oct 19, 2002
Description corrected (changed 'orders' to 'degrees') by Jason Kimberley, Sep 06 2009
Extended to the sixteenth row (in the b-file) by Jason Kimberley, Sep 24 2009
STATUS
approved