

A051031


Triangle E(n,r)= number of not necessarily connected rregular 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
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OFFSET

1,18


COMMENTS

A graph in which every node has r edges is called an rregular graph. The triangle is symmetric because if an nnode graph is rregular, than its complement is (n  1  r)regular and two graphs are isomorphic if and only if their complements are isomorphic.


REFERENCES

M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137146. [From Jason Kimberley, Sep 24 2009]


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 kregular 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, Regular Graph.


FORMULA

E(n,r) = A068934(n,r) + A068933(n,r).


EXAMPLE

a(8, 3) = 6. Edgelists for the 6 3regular 8node 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.


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

Eric W. Weisstein


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 bfile) by Jason Kimberley, Sep 24 2009


STATUS

approved



