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We study a very large family of graphs, the members of which comprisedisjoint paths of cliques with extremal cliques identified. This broadcharacterisation naturally generalises those of various smaller families ofgraphs having well-known chromatic polynomials. We derive a relatively simpleformula for an arbitrary member of the subfamily consisting of those graphswhose constituent clique-paths have at least one trivial extremal clique, anduse this formula to show that the set of all non-integer chromatic roots ofthese graphs is closed under multiplication by natural numbers. A well-knownresult of Sokal then leads to our main result, which is that there exists a setof chromatic roots which is closed under positive integer multiplication inaddition to being dense in the complex plane. Our findings lend considerableweight to a conjecture of Cameron, who has suggested that this closure propertymay be a generic feature of the chromatic polynomial. We also hope that theformula we provide will be of use to those computing with chromaticpolynomials.

A set of chromatic roots which is dense in the complex plane and closed under multiplication by positive integers