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no. 5
Operads on graphs: extending the pre-Lie operad and general construction
Aval, Jean-Christophe1; Samuele, Giraudo2; Karaboghossian, Théo1; Tanasa, Adrian3
1 Université de Bordeaux Bordeaux INP CNRS LaBRI UMR 5800 F-33400 Talence (France)
2 Université du Québec à Montréal LaCIM Montréal H2X 3Y7 (Canada)
3 H. Hulubei Natural Institute for Physics and Nuclear Engineering Magurele (Romania)
Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1313-1348

The overall aim of this paper is to define a structure of graph operads, thus generalizing the celebrated pre-Lie operad on rooted trees. More precisely, we define two operads on multigraphs, and exhibit a non-trivial correspondence between them and the pre-Lie and Kontsevich-Willwacher operads. We study one of these operads in more detail. While its structure is too involved to exhibit a description by generators and relations, we show that it has interesting finitely generated sub-operads, with links with the commutative and the magmatic commutative operads. In particular, one of them is Koszul this allows us to compute its Koszul dual. Finally, we introduce a new framework on species and operads and a general way to define operads on multigraphs.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.442
Classification: 05E99, 05C76, 05C25
Keywords: graphs, species, operads, pre-Lie operad, Koszul duality

Aval, Jean-Christophe 1; Samuele, Giraudo 2; Karaboghossian, Théo 1; Tanasa, Adrian 3

1 Université de Bordeaux Bordeaux INP CNRS LaBRI UMR 5800 F-33400 Talence (France)
2 Université du Québec à Montréal LaCIM Montréal H2X 3Y7 (Canada)
3 H. Hulubei Natural Institute for Physics and Nuclear Engineering Magurele (Romania)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Aval, Jean-Christophe; Samuele, Giraudo; Karaboghossian, Théo; Tanasa, Adrian. Operads on graphs: extending the pre-Lie operad and general construction. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1313-1348. doi: 10.5802/alco.442

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