A129422 -id:A129422

A333330

Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

+10
13

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 3, 2, 1, 1, 1, 0, 1, 0, 4, 0, 4, 0, 1, 1, 0, 1, 1, 5, 7, 9, 4, 1, 1, 1, 0, 1, 0, 7, 0, 24, 0, 7, 0, 1, 1, 0, 1, 1, 8, 16, 54, 60, 32, 8, 1, 1, 1, 0, 1, 0, 10, 0, 128, 0, 240, 0, 12, 0, 1, 1, 0, 1, 1, 12, 37, 271, 955, 1753, 930, 135, 14, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,26

COMMENTS

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333351. Burnside's lemma can be used to extend this method to the unlabeled case.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..350

EXAMPLE

Array begins:

=================================================

n\k | 0 1 2 3 4 5 6 7 8

----+--------------------------------------------

0 | 1 1 1 1 1 1 1 1 1 ...

1 | 1 0 0 0 0 0 0 0 0 ...

2 | 1 1 1 1 1 1 1 1 1 ...

3 | 1 0 1 0 1 0 1 0 1 ...

4 | 1 1 2 3 4 5 7 8 10 ...

5 | 1 0 2 0 7 0 16 0 37 ...

6 | 1 1 4 9 24 54 128 271 582 ...

7 | 1 0 4 0 60 0 955 0 12511 ...

8 | 1 1 7 32 240 1753 13467 90913 543779 ...

9 | 1 0 8 0 930 0 253373 0 35255015 ...

...

CROSSREFS

Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.

Row n=4 is A001399.

Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Mar 15 2020

STATUS

approved

A129416

Number of isomorphism classes of 3-regular loopless multigraphs of order 2n.

+10
9

1, 3, 9, 32, 135, 709, 4637, 38374, 391473, 4764778, 66913591, 1056886475, 18446472265, 351482430368, 7247888726269, 160671989129665, 3808499268504548, 96094161981827499, 2570930535917564366, 72688753062897675445

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

LINKS

Table of n, a(n) for n=1..20.

L. Travis, Graphical Enumeration: A Species-Theoretic Approach, arXiv:math/9811127 [math.CO], 1998.; Ph.D. thesis, Brandeis University, 1999, Section 4.3.

FORMULA

Euler transform of A000421.

CROSSREFS

Column k=3 of A333330.

Cf. A000421 (connected, inv. Eul. trans.), A129427, A129418, A129420, A129422, A129424, A129426.

KEYWORD

nonn

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

a(13)-a(20) from Andrew Howroyd, Mar 19 2020

STATUS

approved

A129421

Number of isomorphism classes of connected 6-regular loopless multigraphs of order n.

+10
9

0, 1, 1, 6, 15, 120, 933, 13303, 252207, 6450828, 205475039, 7936493756, 363639228194, 19476976825809, 1205115679461426, 85288127619421544, 6845235025444882069, 618411485467843477405, 62471139399366989007575, 7014991719815977343879171

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

LINKS

Table of n, a(n) for n=1..20.

Brendan McKay and Adolfo Piperno, Nauty and Traces

FORMULA

Inverse Euler transform of A129422. - Andrew Howroyd, Mar 17 2020

PROG

(nauty/bash) geng -c -d1 ${n} -q | multig -r6 -u # Natan Arie Consigli, Jun 05 2017

CROSSREFS

Column k=6 of A328682.

Cf. A129422, A129432, A000421, A129417, A129419, A129423, A129425, A328682.

KEYWORD

nonn

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

a(1)=0 prepended and a(14)-a(20) from Andrew Howroyd, Mar 17 2020

STATUS

approved

A129426

Number of isomorphism classes of 8-regular loopless multigraphs of order n.

+10
9

0, 1, 1, 10, 37, 582, 12511, 543779, 35255015, 3230979297, 397550237967, 63834143947661, 13080849749829233, 3358751856150607392, 1063851391062768324862, 410060430118305494628648, 190065946515113295597969794, 104826174445642584491349328181

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

LINKS

Table of n, a(n) for n=1..18.

FORMULA

Euler transform of A129425. - Andrew Howroyd, Mar 17 2020

CROSSREFS

Column k=8 of A333330.

Cf. A129425, A129437, A129416, A129418, A129420, A129422, A129424.

KEYWORD

nonn

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

a(1)=0 prepended and a(12)-a(18) from Andrew Howroyd, Mar 17 2020

STATUS

approved

A129418

Number of isomorphism classes of 4-regular loopless multigraphs of order n.

+10
8

1, 0, 1, 1, 4, 7, 24, 60, 240, 930, 4701, 26637, 178569, 1339529, 11187064, 101871881, 1002594996, 10574095327, 118850827173, 1417140114336, 17860018997346, 237160827107408, 3309078044759285, 48396906463199522, 740331404753448181, 11821525310570525197

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Also number of carbon allotropes satisfying the octet rule, excluding stereoisomers. - Natan Arie Consigli, Jun 06 2017

LINKS

Table of n, a(n) for n=0..25.

FORMULA

Euler transform of A129417. - Andrew Howroyd, Mar 14 2020

PROG

(nauty/bash) geng -d1 ${n} -q | multig -r4 -u # Natan Arie Consigli, Jun 06 2017

CROSSREFS

Column k=4 of A333330.

Cf. A129417, A129429, A129416, A129420, A129422, A129424, A129426.

KEYWORD

nonn

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

a(0)-a(1) by Natan Arie Consigli, Jun 06 2017

a(18)-a(25) from Andrew Howroyd, Mar 17 2020

STATUS

approved

A129420

Number of isomorphism classes of 5-regular loopless multigraphs of order 2n.

+10
8

1, 5, 54, 1753, 189341, 46935710, 20494522535, 14041749098602, 14155266802426836, 20061744131278672638, 38587417589460488631726, 97900485588988429336271590, 320012505326477694925887757141, 1321269556386383657509085883067690, 6775074159053505093089897813890701467

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

LINKS

Table of n, a(n) for n=1..15.

FORMULA

Euler transform of A129419. - Andrew Howroyd, Mar 17 2020

CROSSREFS

Column k=5 of A333330.

Cf. A129419, A129431, A129416, A129418, A129422, A129424, A129426.

KEYWORD

nonn

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

a(8)-a(15) from Andrew Howroyd, Mar 21 2020

STATUS

approved

A129424

Number of isomorphism classes of 7-regular loopless multigraphs of order 2n.

+10
8

1, 8, 271, 90913, 165134083, 861889423083, 10352781874872853, 253226977418079364874, 11542716741789258789499297, 915461016322959135074642849828

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

LINKS

Table of n, a(n) for n=1..10.

FORMULA

Euler transform of A129423. - Andrew Howroyd, Mar 21 2020

CROSSREFS

Column k=7 of A333330.

Cf. A129423, A129435, A129416, A129418, A129420, A129422, A129426.

KEYWORD

nonn,more

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

a(7)-a(10) from Andrew Howroyd, Mar 21 2020

STATUS

approved

A129433

Number of isomorphism classes of 6-regular multigraphs of order n, loops allowed.

+10
8

1, 4, 13, 66, 384, 3369, 40365, 675368, 14843787, 412444439, 14024069358, 570883006810, 27368160642418, 1525274952640101, 97766102550401217, 7141331463919539567, 589726897233157151109, 54669618693986578729541, 5653861386413841612952683, 648651469510725736002532451

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

LINKS

Table of n, a(n) for n=1..20.

R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436. [From Jason Kimberley, Oct 05 2009]

FORMULA

a(n) = N\{S_n[S_6] * S_{3n}[S_2]\}. [Equation (5.8) of Read 1959]

Euler transform of A129432. - Andrew Howroyd, Mar 15 2020

CROSSREFS

Column k=6 of A167625.

Cf. A129432, A129422, A129427, A129429, A129431, A129435, A129437.

KEYWORD

nonn

AUTHOR

Brendan McKay, Apr 15 2007

EXTENSIONS

Using the formula, new terms a(13), Oct 05 2009, and a(14)-a(16), Nov 09 2009, were computed in MAGMA by Jason Kimberley

a(17)-a(20) from Andrew Howroyd, Mar 21 2020

STATUS

approved