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The overall aim of this paper is to define a structure of graph operads, thusgeneralizing the celebrated pre-Lie operad on rooted trees. More precisely, wedefine two operads on multigraphs, and exhibit a non trivial link between themand the pre-Lie and Kontsevich- Willwacher operads. We study one of theseoperads in more detail. While its structure is too involved to exhibit adescription by generators and relations, we show that it has interestingfinitely generated sub-operads, with links with the commutative and themagmatic commutative operads. In particular, one of them is Koszul and thisallows us to compute its Koszul dual. Finally, we introduce a new framework onspecies and operads and a general way to define operads on multigraphs.

Operads on graphs: extending the pre-Lie operad and general construction