  We prove that the genus polynomials of the graphs called iterated claws are
real-rooted. This continues our work directed toward the 25-year-old conjecture
that the genus distribution of every graph is log-concave. We have previously
established log-concavity for sequences of graphs constructed by iterative
vertex-amalgamation or iterative edge-amalgamation of graphs that satisfy a
commonly observable condition on their partitioned genus distributions, even
though it had been proved previously that iterative amalgamation does not
always preserve real-rootedness of the genus polynomial of the iterated graph.
In this paper, the iterated topological operations are adding a claw and adding
a 3-cycle, rather than vertex- or edge-amalgamation. Our analysis here
illustrates some advantages of employing a matrix representation of the
transposition of a set of productions.
