cp's OEIS Frontend

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

%I A244742 #34 Jan 08 2025 10:53:51
%S A244742 1,2,3,6,10,19,32,57,94
%N A244742 Dimension of the vector space of 4-invariants on simple unlabeled graphs on n vertices.
%C A244742 An invariant on graphs is a function that takes the same values on isomorphic graphs.
%C A244742 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 A244742 f(G) - f(r(G,A,B)) = f(t(G,A,B)) - f(r(t(G,A,B),A,B)),
%C A244742 where:
%C A244742 r(G,A,B)=r(G,B,A) is a graph obtained from G by removal of edge (A,B);
%C A244742 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 A244742 The 4-invariants on graphs with n vertices form a vector space, whose dimension is given by this sequence.
%C A244742 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 A244742 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 A244742 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 A244742 S. K. Lando, <a href="https://doi.org/10.1007/s10688-006-0001-8">J-invariants of plane curves and framed chord diagrams</a>, Functional Analysis and Its Applications, 40:1 (2006), 1-13.
%Y A244742 Cf. A000088, A245246, A362740.
%K A244742 nonn,hard,more
%O A244742 1,2
%A A244742 _Max Alekseyev_, Jul 05 2014
%E A244742 a(1)-a(7) are given by S.K. Lando.
%E A244742 a(8) from _Max Alekseyev_, Jul 11 2014
%E A244742 a(9) from _Max Alekseyev_, May 08 2023