Iterated Claws Have Real-rooted Genus Polynomials

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 vertexamalgamation 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 realrootedness of the genus polynomial of the iterated graph. In this paper, the iterated topological operation is adding a claw, rather than vertexor edge-amalgamation. Our analysis here illustrates some advantages of employing a matrix representation of the transposition of a set of productions. Version: 10:23 April 6, 2015