摘要：
I will demonstrate a recent result that intersects combinatorics with topology. We prove that the genus polynomials of the graphs called ``iterated claws'' are real-rooted, which implies the log-concavity. This continues a work directed toward the 25-year-old conjecture (M. Furst, J.L. Gross, and R. Statman, Genus distributions for two class of graphs, J. Combin. Theory Ser. B, 46 (1989), 523--534) that the genus distribution of every graph is log-concave. The classical iterative amalgamation operation does not always preserve real-rootedness of the genus polynomial of the iterated graph. In this talk, 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. The proof for the real-rootedness is elemental, based on a 4-term-recurrence relation.