Hopf Algebras of Graphs
Jean-Christophe Novelli (1), Jean-Yves Thibon (1), Nicolas M. Thi\'ery, (2) ((1) Institut Gaspard Monge, Universit\'e de Marne-la-Vall\'ee, France, (2) Laboratoire de Probabilit\'es, Combinatoire et Statistiques, Universit\'e, Claude Bernard Lyon I, France)
TL;DR
This paper introduces graded Hopf algebras constructed from graphs and hypergraphs, generalizing symmetric functions and embedding into polynomial algebras, providing a new algebraic framework for combinatorial graph structures.
Contribution
It defines new Hopf algebras based on graphs and hypergraphs, extending algebraic structures like symmetric functions to combinatorial graph settings.
Findings
Hopf algebras are graded by the number of edges
Algebras embed into polynomial algebras in infinitely many variables
Generalizes symmetric and quasi-symmetric functions
Abstract
We define graded Hopf algebras with bases labeled by various types of graphs and hypergraphs, provided with natural embeddings into an algebra of polynomials in infinitely many variables. These algebras are graded by the number of edges and can be considered as generalizations of symmetric or quasi-symmetric functions.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Molecular spectroscopy and chirality