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We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard matrices of order $64$, we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters $(63,32,16,16)$, $(64,36,20,20)$, and $(64,28,12,12)$ for almost all feasible 2-ranks. In addition we work out the behaviour of the 2-rank for a graph product related to the Kronecker product for Hadamard matrices, which enables us to find many graphical Hadamard matrices of order $4^m$ for which the related strongly regular graphs have an unbounded number of different 2-ranks. The paper extends results from the article Switched symplectic graphs and their 2-ranks by the first and the last author.
In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In particular, we sh
If $q = p^n$ is a prime power, then a $d$-dimensional emph{$q$-Butson Hadamard matrix} $H$ is a $dtimes d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint on the dimens
In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to
Let $n geq r geq s geq 0$ be integers and $mathcal{F}$ a family of $r$-subsets of $[n]$. Let $W_{r,s}^{mathcal{F}}$ be the higher inclusion matrix of the subsets in ${mathcal F}$ vs. the $s$-subsets of $[n]$. When $mathcal{F}$ consists of all $r$-sub
We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are obtained for