We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Szöllősi, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.
Over forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph $X$ contains a Hadamard matrix then $X$ is either of Latin square type or of negative Latin square type. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency algebras. Moreover we show that there are only three distance regular covers of the complete graph that give complex Hadamard matrices.
We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest non-trivial case is M=3, and we obtain here several results, notably with a classification at N=7. We discuss as well the potential applications of the M=3 results to various $M=N$ questions.
In this paper, we find regular or biregular Hadamard matrices with maximum excess by negating some rows and columns of known Hadamard matrices obtained from quadratic residues of finite fields. In particular, we show that if either $4m^2+4m+3$ or $2m^2+2m+1$ is a prime power, then there exists a biregular Hadamard matrix of order $n=4(m^2+m+1)$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to be regular in terms of four-class translation association schemes on finite fields.
Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.
We introduce a class of regular unit Hadamard matrices whose entries consist of two complex numbers and their conjugates for a total of four complex numbers. We then show that these matrices are contained in the Bose-Mesner algebra of an association scheme arising from skew Paley matrices.
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 Hadam…
We consider the regular subgroups of the automorphism group of the linear Hadamard code. These subgroups correspond to the regular subgroups of $GA(r,2)$, w.t.r action on the vectors of $F_2^{r}$, where $n=2^r-1 $ is the length of the Hamadard code. We show that the dihedral group $D_{2^{r-1}}$ is a regular subgroup of $GA(r,2)$ only when $r=3$. Following the approach of \cite{M} we study the regular subgroups of the Hamming code obtained from the regular subgroups of the automorphism group of the Hadamard code of length 15.
In this paper we investigate the structure of the critical groups of doubly regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Reid found the existence of a skew Hadamard matrix of order $n+1$ to be equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices …
It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix of type II. Similarly, any ternary extended quadratic residue code contains the incidence matrix of a Hadamard 3-design associated with a Paley-Hadamard matrix of type I. If $q=5, 11, 17, 23$, then the full permutation automorphism group of $L(q)$ coincides with the full automorphism group of $D(q)$, and a similar result holds for the ternary extended quadr…
A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased in terms of balanced splittable Hadamard matrices, real flat equiangular tight frames, spherical two-distance sets, and two-distance tight frames. We use combinatorial analysis to restrict the parameters of a balanced splittable Hadamard matrix to …
Quantum state transfer within a quantum computer can be achieved by using a network of qubits, and such a network can be modelled mathematically by a graph. Here, we focus on the corresponding Laplacian matrix, and those graphs for which the Laplacian can be diagonalized by a Hadamard matrix. We give a simple eigenvalue characterization for when such a graph has perfect state transfer at time $π/2$; this characterization allows one to choose the correct eigenvalues to build graphs having perfect state transfer. We characterize the graphs that are diagonalizable by the standard Hadamard matrix,…
Free Fermions on vertices of distance-regular graphs are considered. Bipartition are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in details within that framework and the existence of the commuting matrix is …
In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.
Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,λ)$-graphs and symmetric weighing matrices
Hebei Normal University（河北师范大学）；Tilburg University（蒂尔堡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；China Three Gorges University（三峡大学）；Novosibirsk State University（新西伯利亚国立大学）；Jiangxi Normal University（江西师范大学）
Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.
