cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A328682 Array read by antidiagonals: T(n,r) is the number of connected r-regular loopless multigraphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 0, 1, 0, 1, 1, 4, 6, 6, 1, 0, 0, 1, 0, 1, 0, 6, 0, 19, 0, 1, 0, 0, 1, 0, 1, 1, 7, 15, 49, 50, 20, 1, 0, 0, 1, 0, 1, 0, 9, 0, 120, 0, 204, 0, 1, 0, 0, 1, 0, 1, 1, 11, 36, 263, 933, 1689, 832, 91, 1, 0, 0, 1, 0, 1, 0, 13, 0, 571, 0, 13303, 0, 4330, 0, 1, 0, 0, 1, 0, 1, 1, 15, 72, 1149, 12465, 90614, 252207, 187392, 25227, 509, 1, 0, 0
Offset: 0

Views

Author

Natan Arie Consigli, Dec 17 2019

Keywords

Comments

Initial terms computed using 'Nauty and Traces' (see the link).
T(0,r) = 1 because the "nodeless" graph has zero (therefore in this case all) nodes of degree r (for any r).
T(1,0) = 1 because only the empty graph on one node is 0-regular on 1 node.
T(1,r) = 0, for r>0: there's only one node and loops aren't allowed.
T(2,r) = 1, for r>0 since the only edges that are allowed are between the only two nodes.
T(3,r) = parity of r, for r>0. There are no such graphs of odd degree and for an even degree the only multigraph satisfying that condition is the regular triangular multigraph.
T(n,0) = 0, for n>1 because graphs having more than a node of degree zero are disconnected.
T(n,1) = 0, for n>2 since any connected graph with more than two nodes must have a node of degree greater than two.
T(n,2) = 1, for n>1: the only graphs satisfying that condition are the cyclic graphs of order n.
This sequence may be derived from A333330 by inverse Euler transform. - Andrew Howroyd, Mar 15 2020

Examples

			Square matrix T(n,r) begins:
========================================================
n\r | 0     1     2     3     4     5      6      7
----+---------------------------------------------------
  0 | 1,    1,    1,    1,    1,    1,     1,     1, ...
  1 | 1,    0,    0,    0,    0,    0,     0,     0, ...
  2 | 0,    1,    1,    1,    1,    1,     1,     1, ...
  3 | 0,    0,    1,    0,    1,    0,     1,     0, ...
  4 | 0,    0,    1,    2,    3,    4,     6,     7, ...
  5 | 0,    0,    1,    0,    6,    0,    15,     0, ...
  6 | 0,    0,    1,    6,   19,   49,   120,   263, ...
  7 | 0,    0,    1,    0,   50,    0,   933,     0, ...
  8 | 0,    0,    1,   20,  204, 1689, 13303, 90614, ...
  ...

Links

Crossrefs

Rows n=3..9 are: A000035, A253186, A324221, A324218, A324217, A325474, A327604.
Columns r=3..8 are: A000421, A129417, A129419, A129421, A129423, A129425.
Cf. A289986 (main diagonal), A333330 (not necessarily connected), A333397.

Programs

  • nauty
    # This program will execute the "else echo" line if the graph is nontrivial (first three columns, first two rows or both row and column indices are odd)
for ((i=0; i<16; i++)); do
n=0
r=${i}
while ((n<=i)); do
if( (((r==0)) && ((n==0)) ) || ( ((r==0)) && ((n==1)) ) || ( ((r==1)) && ((n==2)) ) || ( ((r==2)) && !((n==1)) ) ); then
echo 1
elif( ((n==0)) || ((n==1)) || ((r==0)) || ((r==1)) || (! ((${r}%2 == 0)) && ! ((${n}%2 == 0)) || ( ((r==2)) && ((n==1)) )) ); then
echo 0
else echo $(./geng -c -d1 ${n} -q | ./multig -m${r} -r${r} -u 2>&1 | cut -d ' ' -f 7 | grep -v '^$');  fi;
((n++))
((r--))
done
done

Formula

Column r is the inverse Euler transform of column r of A333330. - Andrew Howroyd, Mar 15 2020

A324217 Number of connected 2n-regular loopless multigraphs on seven unlabeled nodes.

Original entry on oeis.org

0, 1, 50, 933, 12465, 119906, 864705, 4930446, 23280925, 94308983, 336704397, 1081565510, 3176476119, 8638660038, 21978039571, 52745266132, 120228555981, 261785061087, 547126506745, 1102086250237, 2147107495684
Offset: 0

Views

Author

Natan Arie Consigli, Apr 30 2019

Keywords

Comments

There are no (2n+1)-regular graphs satisfying the condition above.
Multigraphs are loopless.
Initial terms computed with 'nauty and Traces' (see the link).

Links

Crossrefs

Row n=7 of A328682.
Cf. A324218.

Programs

  • nauty
    for ((n=0;n<30;n=n+2)); do geng -c -d1 7 -q | multig -r${n} -u; done

Extensions

a(10)-a(20) from Andrew Howroyd, Mar 17 2020

A325474 Number of connected n-regular loopless multigraphs on eight unlabeled nodes.

Original entry on oeis.org

0, 0, 1, 20, 204, 1689, 13303, 90614, 543116, 2842431, 13218770, 55233883, 210199115, 736166006, 2395665563, 7301022565, 20981692199, 57188601747, 148588072319, 369611918382, 883579452509, 2036672785494, 4539872512172, 9811474182969, 20605911433906, 42141179374995
Offset: 0

Views

Author

Natan Arie Consigli, Aug 09 2019

Keywords

Comments

Multigraphs are loopless.
Initial terms computed using 'Nauty and Traces' (see the link).

Links

Crossrefs

Row n=8 of A328682.
Cf. A324217, A324218.

Programs

  • nauty
    for ((n=0;n<11;n++)); do geng -c -d1 8 -q | multig -r${n} -u; done

Extensions

Terms a(13) and beyond from Andrew Howroyd, Mar 17 2020

A325476 Number of connected regular loopless multigraphs on n unlabeled nodes of degree less than n.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 75, 984, 105831, 35494648, 53493557150, 250087643676776, 4520743153498327817, 272584534800111470995411
Offset: 0

Views

Author

Natan Arie Consigli, May 02 2019

Keywords

Comments

Multigraphs are loopless.
Initial terms computed using nauty and traces.

Examples

			There is no such thing as a graph with nodes of negative degree, and the "nodeless" graph has, according to the definition in the name, zero nodes of degree less than 0. So a(0) = 1.

Links

Crossrefs

Cf. A289988, A289986, A319896, A319897, A324218, A325474, A328682.

Programs

  • nauty
    for ((n=2; n<9; n++)); do
a=0
for ((d=0; d<${n}; d++)); do
s=$(geng -c -d1 ${n} -q | multig -r${d} -u 2>&1 | cut -d ' ' -f 7 | grep -v '^$')
a=$((a+s))
done
echo ${a}
done
# Andrey Zabolotskiy, Sep 26 2019

Formula

a(n) = Sum_{k=0..n-1} A328682(n, k). - Andrew Howroyd, Mar 18 2020

Extensions

a(10)-a(13) from Andrew Howroyd, Mar 18 2020

A327604 Number of connected 2n-regular loopless multigraphs on nine unlabeled nodes.

Original entry on oeis.org

0, 1, 832, 252207, 35241608, 2351580473, 89216898630, 2194653376016, 38376657658125, 509484171323720, 5390386822995890, 47150458233195944, 350827631527584993, 2271025998493100911, 13023793690772493636
Offset: 0

Views

Author

Natan Arie Consigli, Sep 18 2019

Keywords

Comments

There are no (2n+1)-regular graphs satisfying the condition above.
Multigraphs are loopless.
Initial terms computed with 'Nauty and Traces' (see the link).

Links

Crossrefs

Row n=9 of A328682.
Cf. A324217, A324218.

Programs

  • nauty
    for ((n=0;n<12;n=n+2)); do geng -c -d1 9 -q | multig -r${n} -u; done

Extensions

a(6)-a(14) from Andrew Howroyd, Mar 18 2020
Showing 1-5 of 5 results.

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