  A generalized vertex join of a graph is obtained by joining an arbitrary
multiset of its vertices to a new vertex. We present a low-order polynomial
time algorithm for finding the chromatic polynomials of generalized vertex
joins of trees, and by duality we find the flow polynomials of arbitrary
outerplanar graphs. We also present closed formulas for the chromatic and flow
polynomials of vertex joins of cliques and cycles, otherwise known as
"generalized wheel" graphs.
