TY - JOUR

T1 - Log-concavity of genus distributions of ring-like families of graphs

AU - Gross, Jonathan L.

AU - Mansour, Toufik

AU - Tucker, Thomas W.

PY - 2014/11

Y1 - 2014/11

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84904640666&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2014.05.008

DO - 10.1016/j.ejc.2014.05.008

M3 - Article

AN - SCOPUS:84904640666

SN - 0195-6698

VL - 42

SP - 74

EP - 91

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

ER -