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A244742 Dimension of the vector space of 4-invariants on simple unlabeled graphs on n vertices. 1

%I #32 Jul 05 2023 17:05:22

%S 1,2,3,6,10,19,32,57,94

%N Dimension of the vector space of 4-invariants on simple unlabeled graphs on n vertices.

%C An invariant on graphs is a function that takes the same values on isomorphic graphs.

%C A 4-invariant f is an invariant such that for any graph G and any pair of vertices A,B connected by an edge in G,

%C f(G) - f(r(G,A,B)) = f(t(G,A,B)) - f(r(t(G,A,B),A,B)),

%C where:

%C r(G,A,B)=r(G,B,A) is a graph obtained from G by removal of edge (A,B);

%C t(G,A,B) is a graph H obtained from G by modifying the neighborhood of vertex A: N_H(A) is the symmetric difference of N_G(A) and N_G(B). (Note that t(G,A,B) and t(G,B,A) may differ.)

%C The 4-invariants on graphs with n vertices form a vector space, whose dimension is given by this sequence.

%C Similar 4-invariants can be defined on graphs with each vertex A having a label l(A) from the set {0,1} (cf. A362740).

%H Maksim Karev, <a href="https://arxiv.org/abs/2307.00468">On the primitive subspace of Lando framed graph bialgebra</a>, arXiv:2307.00468 [math.CO], 2023.

%H S. K. Lando, <a href="http://www.mccme.ru/mmks/mar08/Lando.pdf">Graph invariants related to knot invatiants</a>. Moscow Mathematical Conference for School Students, 2008. (in Russian)

%H S. K. Lando, <a href="https://doi.org/10.1007/s10688-006-0001-8">J-invariants of plane curves and framed chord diagramsa>, Functional Analysis and Its Applications, 40:1 (2006), 1-13.

%Y Cf. A000088, A245246, A362740.

%K nonn,hard,more

%O 1,2

%A _Max Alekseyev_, Jul 05 2014

%E a(1)-a(7) are given by S.K. Lando.

%E a(8) from _Max Alekseyev_, Jul 11 2014

%E a(9) from _Max Alekseyev_, May 08 2023

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