cp's OEIS Frontend

A362740 Dimension of the vector space of 4-invariants on simple 01-labeled graphs on n vertices.

%I A362740 #20 Jan 08 2025 11:38:57
%S A362740 2,5,11,26,58,131,283
%N A362740 Dimension of the vector space of 4-invariants on simple 01-labeled graphs on n vertices.
%C A362740 In a 01-labeled graph each vertex v has a label l(v) from the set {0, 1}. The 01-labeled graphs on n vertices are in a one-to-one correspondence with the rooted unlabeled graphs on n+1 vertices (cf. A000666).
%C A362740 An invariant is a function that takes the same values on isomorphic 01-labeled graphs. A 4-invariant f is an invariant such that for any 01-labeled graph G and any pair of vertices A,B connected by an edge in G,
%C A362740 f(G) - f(r(G,A,B)) = f(t(G,A,B)) - f(r(t(G,A,B),A,B)),
%C A362740 where:
%C A362740 r(G,A,B) is a graph obtained from G by removing or adding edge (A,B) when it is present or missing in G, respectively;
%C A362740 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); and if l(B)=1, then also by removing the edge (A,B) and inverting the label l(A) in H.
%C A362740 The 4-invariants on 01-labeled graphs on n vertices form a vector space, whose dimension is given by this sequence.
%H A362740 Maksim Karev, The space of framed chord diagrams as a Hopf module, Journal of Knot Theory and Its Ramifications 24:3 (2015), 1550014. <a href="https://doi.org/10.1142/S0218216515500145">doi:10.1142/S0218216515500145</a> Preprint: <a href="https://arxiv.org/abs/1404.0026">arXiv:1404.0026</a> [math.GT]
%H A362740 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 A362740 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 A362740 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 A362740 Cf. A000666, A244742.
%K A362740 hard,more,nonn
%O A362740 1,1
%A A362740 _Max Alekseyev_, May 01 2023
%E A362740 a(1)-a(5) computed by I. A. Dynnikov.