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has been cited by the following article:

• 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.