TITLE:
Hopf Algebra of Labeled Simple Graphs
AUTHORS:
Jiaming Dong, Huilan Li
KEYWORDS:
Hopf Algebra, Labeled Simple Graph, Conjunction Product, Unshuffle Coproduct, Compatibility
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.13 No.1,
January
31,
2023
ABSTRACT: A lot
of combinatorial objects have a natural bialgebra structure. In this paper, we prove
that the vector space spanned by labeled simple graphs is a bialgebra with the
conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra
since it is graded connected. The main conclusions are that the vector space
spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf
algebra and that there is a Hopf homomorphism from permutations to label simple
graphs.