Quasi-balanced weighing matrices, signed strongly regular graphs and association schemes

TL;DR

This paper introduces quasi-balanced weighing matrices and explores their connection to signed strongly regular graphs and association schemes, providing new insights into their structure and relationships.

Contribution

It defines quasi-balanced weighing matrices and demonstrates their role in signing strongly regular graphs and association schemes, revealing novel structural links.

Findings

Introduction of quasi-balanced weighing matrices

Signed strongly regular graphs characterized by these matrices

Presentation of related association schemes

Abstract

A weighing matrix $W$ is quasi-balanced if $|W||W|^\top=|W|^\top|W|$ has at most two off-diagonal entries, where $|W|_{ij}=|W_{ij}|$. A quasi-balanced weighing matrix $W$ signs a strongly regular graph if $|W|$ coincides with its adjacency matrix. Among other things, signed strongly regular graphs and their equivalent association schemes are presented.