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A051031 Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n. 19

%I

%S 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,

%T 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,

%U 0,1,1,1,9,94,1547,7849,7849,1547,94,9,1,1,1,0,10,0,10786,0,367860,0,10786

%N Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n.

%C 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.

%H Jason Kimberley, <a href="/A051031/b051031.txt">Rows 1..16 of A051031 triangle, flattened</a>

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/A051031">Not necessarily connected connected regular graphs (with girth at least 3)</a>

%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/E_k-reg_girth_ge_g_index">Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g</a>

%H Markus Meringer, <a href="http://dx.doi.org/10.1002/(SICI)1097-0118(199902)30:2&lt;137::AID-JGT7&gt;3.0.CO;2-G">Fast generation of regular graphs and construction of cages</a>, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]

%H Markus Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/pdf/pub/FastGenRegGraphJGT.pdf">Fast Generation of Regular Graphs and Construction of Cages</a>, Univ. Bayreuth, 1999

%H Markus Meringer, <a href="http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html">Tables of Regular Graphs</a> [From _Jason Kimberley_, Sep 24 2009]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RegularGraph.html">Regular Graph.</a>

%F T(n,r) = A068934(n,r) + A068933(n,r).

%e T(8,3) = 6. Edge-lists for the 6 3-regular 8-node graphs:

%e Graph 1: 12, 13, 14, 23, 24, 34, 56, 57, 58, 67, 68, 78

%e Graph 2: 12, 13, 14, 24, 34, 26, 37, 56, 57, 58, 68, 78

%e Graph 3: 12, 13, 23, 14, 47, 25, 58, 36, 45, 67, 68, 78

%e Graph 4: 12, 13, 23, 14, 25, 36, 47, 48, 57, 58, 67, 68

%e Graph 5: 12, 13, 24, 34, 15, 26, 37, 48, 56, 57, 68, 78

%e Graph 6: 12, 23, 34, 45, 56, 67, 78, 18, 15, 26, 37, 48.

%e Triangle starts

%e 1;

%e 1, 1,

%e 1, 0, 1,

%e 1, 1, 1, 1,

%e 1, 0, 1, 0, 1,

%e 1, 1, 2, 2, 1, 1,

%e 1, 0, 2, 0, 2, 0, 1,

%e 1, 1, 3, 6, 6, 3, 1, 1,

%e 1, 0, 4, 0, 16, 0, 4, 0, 1,

%e 1, 1, 5, 21, 60, 60, 21, 5, 1, 1,

%e 1, 0, 6, 0, 266, 0, 266, 0, 6, 0, 1,

%e 1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1,

%Y Row sums give A005176.

%Y 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]

%K nonn,tabl

%O 1,18

%A _Eric W. Weisstein_

%E More terms and comments from _David Wasserman_, Feb 22 2002

%E More terms from _Eric W. Weisstein_, Oct 19, 2002

%E Description corrected (changed 'orders' to 'degrees') by _Jason Kimberley_, Sep 06 2009

%E Extended to the sixteenth row (in the b-file) by _Jason Kimberley_, Sep 24 2009

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Last modified February 20 22:51 EST 2019. Contains 320362 sequences. (Running on oeis4.)