The log concavity of two graphical sequences

2025-01-07

AI Summary

This study investigates the log-concavity of two fundamental sequences in graph theory and coding theory: (1) distance distribution sequences centered at a fixed vertex (i.e., connectivity sequences), and (2) multiplicity sequences of eigenvalues in Q-polynomial association schemes. Employing an integrated approach—combining graph spectral theory, algebraic combinatorics, association scheme theory, and P–Q duality analysis—alongside distance distribution analysis, eigenvalue decomposition, and generating function techniques, we establish that the adjacency degree sequence of any graph’s large Cartesian power is log-concave—a first-time result. We strengthen the classical 1978 theorem on distance-regular graphs and derive novel log-concavity criteria for multiplicity sequences under P–Q duality. Finally, we confirm log-concavity for relevant sequences in strongly regular graphs, doubly-weighted codes, and completely regular codes. These results provide structural guarantees and new analytical tools for spectral graph theory and code construction.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Log-concavity
• Connectivity Sequences
• Graph Theory

Innovation

• Log-concavity
• P-Q Duality
• Q-polynomial Association Schemes