A361408 -id:A361408 - cp's OEIS frontend

A321304 Triangle T(n,f): the number of bicolored connected cubic graphs on 2n vertices with f vertices of the first color.

1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 5, 5, 5, 2, 2, 5, 10, 31, 46, 63, 46, 31, 10, 5, 19, 64, 248, 542, 931, 1052, 931, 542, 248, 64, 19, 85, 490, 2382, 7011, 15199, 23405, 27336, 23405, 15199, 7011, 2382, 490, 85, 509, 4595, 27233, 101002, 268675, 523246, 776657, 882321, 776657, 523246, 268675, 101002, 27233, 4595, 509

Offset: 0

R. J. Mathar, Nov 03 2018

These are connected, undirected, simple cubic graphs where each vertex has either the first or the second color. Row n has 2n+1 entries, 0<=f<=2n. The column f=0 (1, 0, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 2, 10, 64, 490...) counts the rooted cubic graphs.

			The triangle starts:
0 vertices:   1;
2 vertices:   0,  0,   0;
4 vertices:   1,  1,   1,   1,   1;
6 vertices:   2,  2,   5,   5,   5,    2,   2;
8 vertices:   5, 10,  31,  46,  63,   46,  31,  10,   5;
10 vertices: 19, 64, 248, 542, 931, 1052, 931, 542, 248, 64, 19;

Andrew Howroyd, Table of n, a(n) for n = 0..440 (rows 0..20)

Columns f=0, 1, 2 are A002851, A361407, A361408.
Row sums are A361403.
Central coefficients are A361406.
Cf. A294783 (bicolored trees), A321305 (signed edges), A361361 (not necessarily connected).

T(n,f) = T(n,2n-f).

Terms a(49) and beyond from Andrew Howroyd, Mar 11 2023

A361407 Number of connected cubic graphs on 2n unlabeled vertices rooted at a vertex.

Original entry on oeis.org

0, 1, 2, 10, 64, 490, 4595, 51063, 657623, 9592204, 155630924, 2771922417, 53673859357, 1121581872170, 25143397213226, 601751140758134, 15310778492310274, 412656423154230159, 11743600063060974656, 351882591907696156959

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=1 of A321304.

Cf. A002851, A005638, A361408, A361410.

G.f.: B(x)/C(x) where B(x) is the g.f. of A361410 and C(x) is the g.f. of A005638.

A361411 Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

Original entry on oeis.org

0, 1, 5, 33, 257, 2443, 27682, 363759, 5405697, 89134360, 1609418390, 31525697245, 665263778962, 15039817276939, 362579178545598, 9284375250749758, 251643492565059981, 7197256536139662143, 216621907269166632361, 6844093745422473471562

Offset: 1

Andrew Howroyd, Mar 11 2023

nonn

Column k=2 of A361361.

Cf. A005638, A361408, A361410.

A361406 Number of bicolored connected cubic graphs on 2n unlabeled vertices with n vertices of each color.

Original entry on oeis.org

1, 0, 1, 5, 63, 1052, 27336, 882321, 34455134, 1558650424, 80016369538, 4589908631503, 290839634055722, 20171917072658395, 1519875854413728667, 123616508830454828043, 10794216583730162449785, 1007179737486515827821590, 100007950522974604304016627, 10529173417583858651114779790, 1171605981584666223513790021758

Offset: 0

Andrew Howroyd, Mar 11 2023

nonn

Adjacent vertices may have the same color.

			a(1) = 1 since the only cubic graph on 4 vertices is the complete graph. The bicolored graphs are indistinguishable whichever 2 vertices are colored in the first color.

Central coefficients of A321304.

Cf. A002851, A361407, A361408.

cp's OEIS Frontend

A321304 Triangle T(n,f): the number of bicolored connected cubic graphs on 2n vertices with f vertices of the first color.

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A361407 Number of connected cubic graphs on 2n unlabeled vertices rooted at a vertex.

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A361411 Number of cubic graphs on 2n unlabeled vertices rooted at a pair of indistinguishable vertices.

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A361406 Number of bicolored connected cubic graphs on 2n unlabeled vertices with n vertices of each color.

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