  We study a very large family of graphs, the members of which comprise
disjoint paths of cliques with extremal cliques identified. This broad
characterisation naturally generalises those of various smaller families of
graphs having well-known chromatic polynomials. We derive a relatively simple
formula for an arbitrary member of the subfamily consisting of those graphs
whose constituent clique-paths have at least one trivial extremal clique, and
use this formula to show that the set of all non-integer chromatic roots of
these graphs is closed under multiplication by natural numbers. A well-known
result of Sokal then leads to our main result, which is that there exists a set
of chromatic roots which is closed under positive integer multiplication in
addition to being dense in the complex plane. Our findings lend considerable
weight to a conjecture of Cameron, who has suggested that this closure property
may be a generic feature of the chromatic polynomial. We also hope that the
formula we provide will be of use to those computing with chromatic
polynomials.
