Quasi-strongly regular graphs on the flags of symmetric designs
Eugenia O'Reilly-Regueiro, Octavio B. Zapata-Fonseca
TL;DR
The article defines two flag-graphs, Γ1(D) for general BIBDs and Γ2(D) for biplanes, and proves Γ1(D) is an almost-quasi-strongly regular graph with parameters dictated by the BIBD’s λ and symmetry. It establishes that design isomorphism is equivalent to flag-graph isomorphism for Γ1(D) and Γ2(D), and provides explicit spectral characterizations: Γ1(D) has a detailed eigenvalue structure depending on (v,b,r,k,λ) and, in the symmetric case, reduces to a QSRG with concrete parameters; Γ2(D) is likewise a QSRG with its own parameterization. The paper also computes spectra for known small biplanes, highlighting instances where the flag-graphs are determined by spectrum and cases where cospectral but non-isomorphic graphs arise, thereby advancing understanding of spectral characterizations in flag-based graph constructions. These results bridge combinatorial design theory with spectral graph theory, offering tools for identifying and distinguishing designs via their associated flag-graphs.
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 $(v,k,λ)$-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.