OFFSET
1,1
COMMENTS
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).
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,
f(G) - f(r(G,A,B)) = f(t(G,A,B)) - f(r(t(G,A,B),A,B)),
where:
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;
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.
The 4-invariants on 01-labeled graphs on n vertices form a vector space, whose dimension is given by this sequence.
LINKS
Maksim Karev, The space of framed chord diagrams as a Hopf module, Journal of Knot Theory and Its Ramifications 24:3 (2015), 1550014. doi:10.1142/S0218216515500145 Preprint: arXiv:1404.0026 [math.GT]
Maksim Karev, On the primitive subspace of Lando framed graph bialgebra, arXiv:2307.00468 [math.CO], 2023.
S. K. Lando, Graph invariants related to knot invatiants. Moscow Mathematical Conference for School Students, 2008. (in Russian)
CROSSREFS
KEYWORD
hard,more,nonn
AUTHOR
Max Alekseyev, May 01 2023
EXTENSIONS
a(1)-a(5) computed by I. A. Dynnikov.
STATUS
approved