The log concavity of two graphical sequences

TL;DR

This work investigates log-concavity properties of two natural graph-theoretic sequences: the valencies along distance shells from a fixed vertex and the eigenvalue multiplicities in association schemes. It proves that large Cartesian powers $G^{\square n}$ have log-concave valencies, and that distance-regular graphs yield a log-concave valency sequence $(v_i)$, with further implications for strongly regular graphs and two-weight/ completely regular codes. Through P–Q duality, the authors show that multiplicities $m_i$ in $Q$-polynomial association schemes are log-concave under a mild monotonicity condition, providing a route to unimodality results. The results, alongside counterexamples in general graphs and constructive CCC methods, delineate the scope of log-concavity in algebraic graph theory and coding theory, highlighting both the reach and limits of these techniques.

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.

Figures (1)

• Figure 1: TriangleReplacedPetersen Graph

Theorems & Definitions (13)

• Theorem 1
• Example 1
• Example 2
• Remark 1
• Remark 2
• Example 3
• Remark 3
• Example 4
• Theorem 2
• Corollary 1