Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,\lambda)$-graphs and symmetric weighing matrices

TL;DR

This paper explores the relationships between certain combinatorial structures like graphs and matrices, introducing new recursive constructions of Hadamard matrices and identifying fixed-point free involutions in symplectic graphs, advancing the theory of symmetric designs.

Contribution

It presents two new recursive methods for constructing regular symmetric Hadamard matrices and identifies fixed-point free involutions in symplectic graphs, linking graph theory and matrix theory.

Findings

New recursive constructions of Hadamard matrices

Identification of fixed-point free involutions in symplectic graphs

Orthogonal signings for infinite families of antipodal distance-regular graphs

Abstract

In this paper, we illustrate important aspects of the interplay between weighing matrices, $(v,k,\lambda)$-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.