A362740 - OEIS

%I #18 Jul 05 2023 17:05:11

%S 2,5,11,26,58,131,283

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

%C 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 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 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) is a graph obtained from G by removing or adding edge (A,B) when it is present or missing in G, respectively;

%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); and if l(B)=1, then also by removing the edge (A,B) and inverting the label l(A) in H.

%C The 4-invariants on 01-labeled graphs on n vertices form a vector space, whose dimension is given by this sequence.

%H 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 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. A000666, A244742.

%K hard,more,nonn

%O 1,1

%A _Max Alekseyev_, May 01 2023

%E a(1)-a(5) computed by I. A. Dynnikov.