The log concavity of two graphical sequences

TL;DR

This paper proves that large Cartesian powers of graphs and certain graph-related sequences exhibit log-concavity, with implications for graph theory, coding theory, and association schemes.

Contribution

It establishes log-concavity of valencies in Cartesian powers and distance-regular graphs, improving previous results and linking to association schemes.

Findings

Log-concavity of large Cartesian powers of graphs.

Log-concavity of valencies in distance-regular graphs.

Log-concavity of multiplicities in Q-polynomial association schemes.

Abstract

We show that the large Cartesian powers of any graph have log-concave valencies with respect to a ffxed vertex. We show that the series of valencies of distance regular graphs is log-concave, thus improving on a result of (Taylor, Levingston, 1978). Consequences for strongly regular graphs, two-weight codes, and completely regular codes are derived. By P-Q duality of association schemes the series of multiplicities of Q-polynomial association schemes is shown, under some assumption, to be log-concave.