Hadi Kharaghani

Divisible design graphs

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v,k,@l)-graphs, and like (v,k,@l)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as (v,k,@l)-graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.

The excess of complex Hadamard matrices

AbstractA complex Hadamard matrix,C, of ordern has elements 1, −1,i, −i and satisfiesCC*=nInwhereC* denotes the conjugate transpose ofC. LetC=[cij] be a complex Hadamard matrix of order

Mutually unbiased weighing matrices

n. S(C) = \sum\limits_{ij} {c_{ij} }

On unit weighing matrices with small weight

is called the sum ofC. σ(C)=|S(C)| is called the excess ofC. We study the excess of complex Hadamard matrices. As an application many real Hadamard matrices of large and maximal excess are obtained.

Linked systems of symmetric group divisible designs

We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is attained for n = 1, 5, 9. Furthermore, we prove that certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. As a consequence we generate a new pair of unbiased real Hadamard matrices of order 36.

A strongly regular decomposition of the complete graph and its association scheme

Inspired by the many applications of mutually unbiased Hadamard matrices, we study mutually unbiased weighing matrices. These matrices are studied for small orders and weights in both the real and complex setting. Our results make use of and examine the sharpness of a very important existing upper bound for the number of mutually unbiased weighing matrices.

Turyn-Type Sequences: Classification, Enumeration, and Construction

A class of unbiased (−1, 1)-matrices extracted from a single Hadamard matrix is shown to provide uniform imprimitive association schemes of four class and six class.

Non-commutative association schemes and their fusion association schemes

Abstract The structure of unit weighing matrices of order n and weights 2, 3 and 4 is studied. We show that the number of inequivalent unit weighing matrices U W ( n , 4 ) depends on the number of decomposition of n into sums of non-negative multiples of some specific positive integers. Two interesting sporadic cases are presented in order to demonstrate the complexities involved in the classification of weights larger than 4.