{"pk":64970,"title":"Feynman symmetries of the Martin and \\(c_2\\) invariants of regular graphs","subtitle":null,"abstract":"For every regular graph, we define a sequence of integers, using the recursion of the Martin polynomial. We prove that this sequence counts spanning tree partitions and thus constitutes the diagonal coefficients of powers of the Kirchhoff polynomial. We also prove that this sequence respects all known symmetries of Feynman period integrals in quantum field theory. We show that other quantities with this property, the \\(c_2\\) invariant and the extended graph permanent, are essentially determined by our new sequence. This proves the completion conjecture for the \\(c_2\\) invariant at all primes, and also that it is fixed under twists. We conjecture that our invariant is perfect: Two Feynman periods are equal, if and only if, their Martin sequences are equal.\n \nMathematics Subject Classifications: 81Q30, 05C70, 05C45\n \nKeywords: Martin polynomial, transitions, spanning trees, point counts, Feynman integrals, integer sequences, permanent, Prüfer sequence","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Martin polynomial"},{"word":"transitions"},{"word":"spanning trees"},{"word":"point counts"},{"word":"Feynman integrals"},{"word":"integer sequences"},{"word":"permanent"},{"word":"Prüfer sequence"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/06x4w2zp","frozenauthors":[{"first_name":"Erik","middle_name":"","last_name":"Panzer","name_suffix":"","institution":"Mathematical Institute, University of Oxford, Oxford, U.K.","department":""},{"first_name":"Karen","middle_name":"","last_name":"Yeats","name_suffix":"","institution":"Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada","department":""}],"date_submitted":"2025-03-14T16:44:27Z","date_accepted":"2025-03-14T16:44:27Z","date_published":"2025-03-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64970/galley/49780/download/"}]}