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A051031 Triangle 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.

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]

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, 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.

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 b-file) by Jason Kimberley, Sep 24 2009

STATUS

approved

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Last modified July 31 21:33 EDT 2015. Contains 260167 sequences.