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A051031
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Triangle E(n,r)= number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n.
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18
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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; internal format)
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OFFSET
| 1,18
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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.
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REFERENCES
| M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146. [From Jason Kimberley, Sep 24 2009]
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LINKS
| Jason Kimberley, Rows 1..16 of A051031 triangle, flattened
Jason Kimberley, Not necessarily connected connected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Markus Meringer, Tables of Regular Graphs [From Jason Kimberley, Sep 24 2009]
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| E(n,r) = A068934(n,r) + A068933(n,r).
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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.
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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: A123550 A204433 A004578 * A181059 A111915 A066520
Adjacent sequences: A051028 A051029 A051030 * A051032 A051033 A051034
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KEYWORD
| nonn,tabl
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| More terms and comments from David Wasserman (dwasserm(AT)earthlink.net), Feb 22 2002
More terms from Eric Weisstein (eric(AT)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
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