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The number of connected weighted cubic graphs with weight n on 8 vertices.
1

%I #5 Sep 05 2022 19:20:26

%S 5,10,41,98,257,537,1131,2116,3893,6665,11177,17867,28011,42419,63145,

%T 91586,130870,183230,253265,344373,463073,614332,807138,1048517,

%U 1350574,1722948,2181614,2739523,3417356,4232137

%N The number of connected weighted cubic graphs with weight n on 8 vertices.

%C Each vertex of the 5 simple cubic graphs is assigned an integer number (weight) >=1. The weight of the graph is the sum of the weights of the vertices.

%F G.f.: x^8*(x^18 +10*x^16 +5*x^15 +37*x^14 +8*x^13 +75*x^12 +16*x^11 +103*x^10 +16*x^9 +108*x^8 +13*x^7 +86*x^6 +3*x^5 +50*x^ 4+21*x^2 -5*x +5)/((-1+x)^8* (1+x)^4 *(x^2+x+1)^2 *(x^2-x+1) *(1+x^2)^2 *(1+x^4)).

%e a(8)=5 because there are 5 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), A321306 (6 vertices).

%K nonn,easy

%O 8,1

%A _R. J. Mathar_, Nov 03 2018