Extending valency log-concavity beyond distance-regular graphs

Identify a larger class of graphs, beyond distance-regular graphs, for which the sequence of distance valencies is log-concave.

Background

The paper proves that the distance-valency sequence of every distance-regular graph is log-concave. It also shows that log-concavity fails for general graphs and even for some distance-degree-regular graphs, so the theorem cannot be extended indiscriminately to broader graph classes. The authors therefore leave open the task of finding a broader class of graphs in which distance-valency log-concavity remains valid.