Log-concavity of genus distributions of ring-like families of graphs

We calculate genus distribution formulas for several families of ring-like graphs and prove that they are log-concave. The graphs in each of our ring-like families are obtained by applying the self-bar-amalgamation operation to the graphs in a linear family (linear in the sense of Stahl). That is, we join the two root-vertices of each graph in the linear family. Although log-concavity has been proved for many linear families of graphs, the only other ring-like sequence of graphs of rising maximum genus known to have log-concave genus distributions is the recently reinvestigated sequence of Ringel ladders. These new log-concavity results are further experimental evidence in support of the long-standing conjecture that the genus distribution of every graph is log-concave. Further evidence in support of the general conjecture is the proof herein that each partial genus distribution, relative to face-boundary walk incidence on root vertices, of an iterative bar-amalgamations of copies of various given graphs is log-concave, which is an unprecedented result for partitioned genus distributions. Our results are achieved via introduction of the concept of a vectorized production matrix, which seems likely to prove a highly useful operator in the theory of genus distributions and via a new general result on log-concavity.