A321306 The number of connected weighted cubic graphs with weight n on 6 vertices.

2, 2, 7, 12, 26, 41, 76, 113, 183, 264, 393, 543, 768, 1024, 1385, 1801, 2355, 2989, 3811, 4740, 5911, 7234, 8857, 10680, 12883, 15336, 18254, 21496, 25293, 29491, 34361, 39713, 45860, 52598, 60260, 68627, 78079, 88354, 99882, 112385, 126316, 141379, 158082, 176080

Offset

6,1

Comments

Each vertex of the 2 simple cubic graphs is assigned an integer number (weight) >=1. The weight of the graph is the sum of the weights of the vertices.

The cycle indices of the permutation group of vertex permutations of the two cubic graphs on 6 vertices are ( +t[1]^6 +3*t[1]^2*t[2]^2 +2*t[3]^2 +4*t[2]^3 +2*t[6])/12 and +( +t[1]^6 +6*t[1]^4*t[2] +9*t[1]^2*t[2]^2 +4*t[1]^3*t[3] +12*t[1]*t[2]*t[3] +6*t[2]^3 +18*t[2]*t[4] +12*t[6] +4*t[3]^2)/72 . The ordinary generating function of the sequence is obtained by adding the two cycle indices and setting t[i] -> x^i/(1-x^i).

Links

Andrew Howroyd, Table of n, a(n) for n = 6..1000

Formula

G.f.: (x^10 +3*x^8 -x^7 +4*x^6 +4*x^4 +3*x^2 -2*x+2) *x^6/((-1+x)^6 *(1+x)^3 *(1+x^2) *(x^2+x+1)^2 *(x^2-x+1)).

Example

a(6)=2 because there are 2 cubic graphs (see A002851), and if the weight is the same as the number of vertices, there is one case for each.

Crossrefs

Cf. A026810 (4 vertices), A321307 (8 vertices), A005513.

Sequence in context: A320168 A123209 A123604 * A024309 A047756 A047767

Adjacent sequences: A321303 A321304 A321305 * A321307 A321308 A321309

Keyword

nonn,easy

Author

R. J. Mathar, Nov 03 2018

Extensions

Terms a(36) and beyond from Andrew Howroyd, Apr 27 2020

Status

approved