A combinatorial non-commutative Hopf algebra of graphs
G. H. E. Duchamp, L. Foissy, N. Hoang-Nghia, D. Manchon, A. Tanasa
TL;DR
This paper introduces a novel non-commutative Hopf algebra structure for graphs, inspired by quantum field theory concepts like discrete energy scales on edges, extending previous algebraic frameworks for rooted trees.
Contribution
It proposes the first non-commutative Hopf algebra for graphs using QFT-inspired scale assignments, expanding algebraic tools for graph analysis.
Findings
Defines a non-commutative product for graphs
Establishes a Hopf algebra structure for graphs
Connects algebraic structures with quantum field theory ideas
Abstract
A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators).
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Taxonomy
TopicsGraph theory and applications · Advanced Combinatorial Mathematics · Algebraic structures and combinatorial models