Quasi-strongly regular graphs on the flags of symmetric designs

TL;DR

This paper explores the properties of flag-graphs derived from symmetric designs, proving they are quasi-strongly regular and examining their spectral determination for known biplanes.

Contribution

It introduces a new perspective on flag-graphs of symmetric designs, proving their quasi-strongly regularity and extending spectral analysis to biplanes.

Findings

Flag-graphs of symmetric designs are quasi-strongly regular.

Flag-graphs of biplanes are also quasi-strongly regular with specific parameters.

Spectral determination of these graphs varies among known biplanes.

Abstract

This paper was inspired by a paper by Blokhuis and Brouwer [Designs, Codes and Cryptography 65, 2012] in which a definition of a graph on the flags of a biplane is given, and they prove that the graph corresponding to the unique $(11,5,2)$-biplane is determined by its spectrum. It is also inspired by the different definition of flag-graph seen in the context of maps and abstract polytopes. Here we use this definition for $(v,k,\lambda)$-BIBDs, and prove that if the design is symmetric then the graph is quasi-strongly regular. We will also use the definition given by Blokhuis and Brouwer for the case of biplanes and prove that this too, is a QSRG, (with different parameters). We investigate whether these graphs are determined by their spectra for some of the known biplanes.