Characterization Tensors of Balanced Incomplete Block Designs

TL;DR

This paper introduces a novel tensor-based characterization method for balanced incomplete block designs (BIBDs), enabling new spectral analysis tools to study their properties across various fields.

Contribution

It constructs strongly symmetric tensors for BIBDs and explores their spectral properties, providing a new mathematical framework for analyzing BIBDs.

Findings

Spectral properties of characterization tensors are established.

Signless characterization tensors offer additional insights.

The approach provides a new tool for studying BIBDs.

Abstract

Balanced incomplete block designs (BIBDs) have wide applications in engineering, business and sciences. In this paper, for each (v, k, \lambda)-BIBD, we construct a strongly symmetric k-th order v-dimensional tensor. We call such a strongly symmetric tensor the characterization tensor of that BIBD, and the absolute value tensor of the characterization tensor the signless characterization tensor of that BIBD. We study some spectral properties of such characterization tensors and signless characterization tensors. In this way, we provide a new tool to study BIBDs.