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Inspired by the foliage of a graph, our invariant has a natural graphical representation in terms of leaves, axils, and twins. It captures both, the connection structure of a graph and the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>2<\/mml:mn><\/mml:math>-body marginal properties of the associated graph state. We relate the foliage partition to the size of LC-orbits and use it to bound the number of LC-automorphisms of graphs. We also show the invariance of the foliage partition when generalized to weighted graphs and qudit graph states.<\/jats:p>","DOI":"10.22331\/q-2025-04-24-1720","type":"journal-article","created":{"date-parts":[[2025,4,24]],"date-time":"2025-04-24T14:00:36Z","timestamp":1745503236000},"page":"1720","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0418-257X","authenticated-orcid":false,"given":"Adam","family":"Burchardt","sequence":"first","affiliation":[{"name":"QuSoft, CWI and University of Amsterdam, Science Park 123, 1098 XG Amsterdam, the Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9349-4075","authenticated-orcid":false,"given":"Frederik","family":"Hahn","sequence":"additional","affiliation":[{"name":"Electrical Engineering and Computer Science Department, Technische Universit\u00e4t Berlin, 10587 Berlin, Germany"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2025,4,24]]},"reference":[{"doi-asserted-by":"publisher","unstructured":"Daniel Gottesman. 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