Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,λ)$-graphs and symmetric weighing matrices
Sergey Goryainov, Willem H. Haemers, Elena V. Konstantinova, Honghai Li
TL;DR
The paper explores the interplay between thin divisible design graphs, fixed-point free involutions, weighing matrices, and signed graphs with orthogonal adjacency matrices. It develops two recursive constructions of regular symmetric Hadamard matrices with constant diagonal (RSHCD) and uses them to build infinite families of thin DDGs via complete multipartite graphs, while also showing that Sp(4,q) (q odd) admits a fixed-point free involution leading to orthogonal signings for an infinite family of antipodal diameter-3 distance-regular graphs (Mathon graphs). By linking (v,k,λ)-graphs with FP-involutions to thin DDGs through an R,Q pair framework, the work bridges design theory, matrix theory, and distance-regular graphs to produce new families of SRGs and orthogonal signings. The results have implications for constructing signed graphs with constrained eigenstructure and for understanding the combinatorial structure underlying symmetric design-inspired objects.
Abstract
In this paper, we illustrate important aspects of the interplay between weighing matrices, $(v,k,λ)$-graphs with fixed-point free involutions, and signed graphs with an orthogonal adjacency matrix, which arises from thin divisible design graphs. In particular, we present two new recursive constructions of regular symmetric Hadamard matrices with constant diagonal (equivalently, two new recursive constructions of strongly regular graphs) and we find a fixed-point free involution in the symplectic graph $Sp(4,q)$, where $q$ is odd, which leads to orthogonal signings for an infinite family of antipodal distance-regular graphs of diameter 3.