Iterated claws have real-rooted genus polynomials - Gross, Jonathan L. (Ars mathematica contemporanea, 2016)
We prove that the genus polynomials of the graphs called iterated claws are realrooted. 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 edgeamalgamation 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 operation is adding a claw, 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.
Keywords: graph genus polynomials; log-concavity; log-konkavnost; realni koreni; real-rootedness; rodovni polinomi grafa; topological graph theory; topološka teorija grafov