{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T16:07:28Z","timestamp":1759939648622,"version":"3.41.0"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2007,5,1]],"date-time":"2007-05-01T00:00:00Z","timestamp":1177977600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2007,5]]},"abstract":"<jats:p>The chromatic polynomial <jats:italic>P<\/jats:italic>\u0393(<jats:italic>x<\/jats:italic>) of a graph \u0393 is a polynomial whose value at the positive integer <jats:italic>k<\/jats:italic> is the number of proper <jats:italic>k<\/jats:italic>-colourings of \u0393. If <jats:italic>G<\/jats:italic> is a group of automorphisms of \u0393, then there is a polynomial OP<jats:sub>\u0393,<jats:italic>G<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>), whose value at the positive integer <jats:italic>k<\/jats:italic> is the number of orbits of <jats:italic>G<\/jats:italic> on proper <jats:italic>k<\/jats:italic>-colourings of \u0393.<\/jats:p>\n\t  <jats:p>It is known that real chromatic roots cannot be negative, but they are dense in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306008200_inline1\">\n\t      <jats:alt-text>$\\lsqb \\frac{32}{27},$<\/jats:alt-text>\n\t    <\/jats:inline-graphic> \u221e). Here we discuss the location of real orbital chromatic roots. We show, for example, that they are dense in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548306008200_inline2\">\n\t      <jats:alt-text>$\\mathbb{R}$<\/jats:alt-text>\n\t    <\/jats:inline-graphic>, but under certain hypotheses, there are zero-free regions.<\/jats:p>\n\t  <jats:p>We also look at orbital flow roots. Here things are more complicated because the orbit count is given by a multivariate polynomial; but it has a natural univariate specialization, and we show that the roots of these polynomials are dense in the negative real axis.<\/jats:p>","DOI":"10.1017\/s0963548306008200","type":"journal-article","created":{"date-parts":[[2006,10,26]],"date-time":"2006-10-26T07:49:21Z","timestamp":1161848961000},"page":"401-407","source":"Crossref","is-referenced-by-count":5,"title":["Orbital Chromatic and Flow Roots"],"prefix":"10.1017","volume":"16","author":[{"given":"PETER J.","family":"CAMERON","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K. K.","family":"KAYIBI","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2007,5,1]]},"reference":[{"key":"S0963548306008200_manual_ref-1","unstructured":"[1] Cameron, P. J. , Jackson, B. and Rudd, J. Orbit-counting polynomials for graphs and codes. Discrete Math., submitted."},{"key":"S0963548306008200_manual_ref-5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397003131"},{"key":"S0963548306008200_manual_ref-3","doi-asserted-by":"publisher","DOI":"10.1007\/s00022-003-1694-y"},{"volume-title":"Chromatic polynomials","year":"1994","author":"Wakelin","key":"S0963548306008200_manual_ref-7"},{"key":"S0963548306008200_manual_ref-2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300000705"},{"key":"S0963548306008200_manual_ref-6","first-page":"474","article-title":"On the imbedding of linear graphs in surfaces","volume":"51","author":"Tutte","year":"1950","journal-title":"Proc. London Math. Soc."},{"key":"S0963548306008200_manual_ref-4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548303006023"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548306008200","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T03:29:59Z","timestamp":1750476599000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548306008200\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,5]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["S0963548306008200"],"URL":"https:\/\/doi.org\/10.1017\/s0963548306008200","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2007,5]]}}}