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 A051031 Triangle read by rows: E(n,r)= number of  not necessarily connected r-regular graphs with n nodes, 0 <= r < n. 19
 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, 1, 0, 10, 0, 10786, 0, 367860, 0, 10786 (list; table; graph; refs; listen; history; text; internal format)
 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. LINKS Jason Kimberley, Rows 1..16 of A051031 triangle, flattened 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, Tables of Regular Graphs [From Jason Kimberley, Sep 24 2009] Eric Weisstein's World of Mathematics, Regular Graph. FORMULA E(n,r) = A068934(n,r) + A068933(n,r). EXAMPLE a(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 Row sums give A005176. Derived from the aforementioned symmetry and the following sequences that count regular graphs of degree k: A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7). [From Jason Kimberley, Sep 24 2009] Sequence in context: A204433 A004578 A234514 * A181059 A111915 A066520 Adjacent sequences:  A051028 A051029 A051030 * A051032 A051033 A051034 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

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