The 224 non-chordal graphs on less than 10 vertices whose chromatic polynomials have no complex roots

We show that all non-chordal graphs up to 9 vertices whose chromatic polynomials have no complex roots can be generated by applying a (sequence of) few operations to a small clique. These operations are the chromatic expansion (connecting a new vertex to every vertex of the given graph), the pyramid (connecting a new vertex to a clique of the given graph), and two new operations - here called bipyramid and tripyramid - that generalize the previous one. We also exhibit the smallest non-chordal graph whose chromatic polynomial has only integer and irrational roots, the smallest graph with a chordless circuit of length 5 whose chromatic polynomial has no complex roots and introduce some infinite families of non-chordal graphs whose chromatic polynomials have no complex roots.