A block negacyclic Bush-type Hadamard matrix and two strongly regular graphs
A Bush-type Hadamard matrix \(H= (H_{ij})\) is a block matrix of order \(4n^2\) with blocks of sizes \(2n\) such that the row and column sums are \(n\) and \(H_{ii}= J_{2n}\), \(H_{ij} J_{2n}= J_{2n} H_{ij}= 0\), for \(1\leq i\neq j\leq 2n\), where \(J_m\) is a \(m\times m\) matrix of all ones. If \(U\) is a circulant matrix of order \(n\) with first row \((0\;1\;0\;0\dots 0)\) and \(D\) a diagonal matrix of order \(n\) with diagonal elements \(-1,1,1,\dots\), then matrices which are polynomials of \(\pi= UD\) are called negacyclic. In this paper a block negacyclic Bush-type Hadamard matrix of order 36 is constructed and using it in two balanaced generalized weighing matrices, twin strongly regular graphs are constructed. A twin graph is a \((0,\pm 1)\)-matrix where each sign \(1\) or \(-1\) provides the adjacency matrix of the graph.
scientific article; zbMATH DE number 1760671
scientific article