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The number of connected weighted cubic graphs with weight n on 6 vertices.
3

%I #8 Apr 27 2020 14:47:33

%S 2,2,7,12,26,41,76,113,183,264,393,543,768,1024,1385,1801,2355,2989,

%T 3811,4740,5911,7234,8857,10680,12883,15336,18254,21496,25293,29491,

%U 34361,39713,45860,52598,60260,68627,78079,88354,99882,112385,126316,141379,158082,176080

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

%C 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.

%C 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).

%H Andrew Howroyd, <a href="/A321306/b321306.txt">Table of n, a(n) for n = 6..1000</a>

%F 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)).

%e 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.

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

%K nonn,easy

%O 6,1

%A _R. J. Mathar_, Nov 03 2018

%E Terms a(36) and beyond from _Andrew Howroyd_, Apr 27 2020