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### Combinatorial equivalence of (0, 1) circulant matrices

Melvin A. Breuer
1969 Journal of computer and system sciences (Print)
In this paper various properties of (0, 1) square matrices are investigated, and in particular, circulant matrices are considered.  ...  If r is a (0, 1) vector or code word of length n and contains exactly K one elements, then let A(r) be the corresponding circulant matrix having r as its first row.  ...  BACKGROUND Tucker [5] has formulated the following concept of combinatorial equivalence of matrices.  ...

### Circulant weighing matrices

Krishnasamy Thiru Arasu, Alex J. Gutman
2010 Cryptography and Communications
We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants.  ...  In particular, we construct W (148, 144), W (152, 144), W (156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs.  ...  The above definition of disjoint support for arbitrary matrices (i.e. not necessarily circulant) boils down to the Definition 4 of disjoint support for circulant matrices.  ...

### Graphs with circulant adjacency matrices

Bernard Elspas, James Turner
1970 Journal of Combinatorial Theory
Properties of a graph (directed or undirected) whose adjacency matrix is a circulant are studied.  ...  Two different criteria are given under which two graphs with circulant adjacency matrices are isomorphic if and only if their connection sets are multiplicatively related.  ...  OPEN PROBLEMS The theorems of this paper have given only a partial description of pairs of graphs with circulant adjacency matrices for which isomorphism is equivalent to having equivalent connection sets  ...

### Circulant q-Butson Hadamard matrices [article]

Trevor Hyde, Joseph Kraisler
2017 arXiv   pre-print
We use algebraic number theory to prove a strong constraint on the dimension of a circulant q-Butson Hadamard matrix when d = p^m and then explicitly construct a family of examples in all possible dimensions  ...  These results relate to the long-standing circulant Hadamard matrix conjecture in combinatorics.  ...  Circulant q-BH matrices of dimension d are equivalent to δ-fibrous functions f : Z/(d) → Z/(q). Theorem 3. Let q = p n be a prime power and ζ a primitive qth root of unity.  ...

### Factoring Matrices into the Product of Circulant and Diagonal Matrices

Marko Huhtanen, Allan Perämäki
2015 Journal of Fourier Analysis and Applications
A generic matrix A ∈ C n×n is shown to be the product of circulant and diagonal matrices with the number of factors being 2n − 1 at most.  ...  For the linear factors, the sum of two PD matrices is factored into the product of two diagonal matrices and a circulant matrix.  ...  Although the notion of permutation equivalence is graph theoretically nonstandard, combinatorial linear algebraically it is perfectly natural [2, p. 4 ].  ...

### The inverses of some circulant matrices

A. Carmona, A.M. Encinas, S. Gago, M.J. Jiménez, M. Mitjana
2015 Applied Mathematics and Computation
In particular, we recover the inverses of some well known circulant matrices whose coefficients are arithmetic or geometric sequences, Horadam numbers among others.  ...  We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse.  ...  Notice that Circ b(2, −1, 0) is nothing but the so called combinatorial Laplacian of a n-cycle.  ...

### Circulant matrices and mathematical juggling

Richard A. Brualdi, Michael W. Schroeder
2018 The Art of Discrete and Applied Mathematics
Circulants form a well-studied and important class of matrices, and they arise in many algebraic and combinatorial contexts, in particular as multiplication tables of cyclic groups and as special classes  ...  Schroeder) 1 If t i = 0, then there is no ball to toss at time i. c b This work is licensed under http://creativecommons.org/licenses/by/3.0/  ...  In Section 3, we elaborate on the connection between juggling sequences and circulant matrices as discussed in [3] , and relate juggling sequences to the permanent of circulants defined in terms of n  ...

### The Binary Rank of Circulant Block Matrices [article]

Ishay Haviv, Michal Parnas
2022 arXiv   pre-print
rank by no more than the maximum of (n_i,k_i)-1 over all i ∈ [m].  ...  Using this method, we prove that the binary rank of the complement of a (k_1, ..., k_m ; n_1, ..., n_m) circulant block diagonal matrix for integers satisfying n_i>k_i>0 for each i ∈ [m] exceeds its real  ...  Equivalently, it is the smallest size of a partition of the ones in M into monochromatic combinatorial rectangles.  ...

### Small circulant complex Hadamard matrices of Butson type [article]

Gaurush Hiranandani, Jean-Marc Schlenker
2014 arXiv   pre-print
We study the circulant complex Hadamard matrices of order n whose entries are l-th roots of unity.  ...  We then provide a list of equivalence classes of such matrices, for small values of n,l.  ...  Shortly after we posted a question 2 which is a more elementary but equivalent statement of Theorem 5.1, Noam Elkies posted an answer to this question which is basically a proof of Theorem 5.1.  ...

### The inverse matrix of some circulant matrices [article]

A. Carmona, A.M. Encinas, S. Gago, M.J. Jiménez, M. Mitjana
2015 arXiv   pre-print
In particular, we recover the inverses of some well known circulant matrices whose coeffifficients are arithmetic or geometric sequences, Horadam numbers among others.  ...  We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse.  ...  Notice that Circ b(2, −1, 0) is nothing but the so called combinatorial Laplacian of a n-cycle.  ...

### Circulant weighing matrices of weight 22t

K. T. Arasu, Ka Hin Leung, Siu Lun Ma, Ali Nabavi, D. K. Ray-Chaudhuri
2006 Designs, Codes and Cryptography
In this paper, we prove the nonexistence of two weighing matrices of weight 81, namely CW (88, 81) and CW (99, 81).  ...  By using zero-based indexing, we define n − 1} and N = {g i |W 0,i = −1, i = 0, 1 . . . , n − 1} , MSC(2010): Primary: 05C15; Secondary: 20D60.  ...  A table of these matrices follows below along with the parameter values that would satisfy the hypothesis of Theorem 4.3.  ...

### Circulant matrices and Galois-Togliatti systems [article]

Pietro De Poi, Emilia Mezzetti, Mateusz Michałek, Rosa Maria Miró-Roig, Eran Nevo
2018 arXiv   pre-print
The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix.  ...  As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems.  ...  In this article, we are interested in the so-called r-lines circulant matrices, i.e. circulant matrices Circ(x 0 , x 1 . . . , x d−1 ) of order d > r, where d−r among x 0 , . . . , x d−1 are specialized  ...

### The geometry of an interchange: Minimal matrices and circulants

1997 Linear Algebra and its Applications
As a by-product, the interchange distance between a special circulant and the set of minimal matrices in its class is determined. Several open problems are stated.  ...  A general method is given for constructing minimal matrices using circulants.  ...  INTRODUCTION Two classes of combinatorial matrices are of special interest in what follows.  ...

### Partial geometric designs having circulant concurrence matrices [article]

Sung-Yell Song, Theodore Tranel
2022 arXiv   pre-print
A partial geometry has two concurrences 1 and 0 and a transversal design TD_λ(k, u) has two concurrences λ and 0.  ...  In this paper, we show the existence of other partial geometric designs having two or three distinct concurrences, and investigate which symmetric circulant matrices are realized as the concurrence matrices  ...  PGDs having circulant concurrence matrices We notice that the concurrence matrices of many PGDs are circulant.  ...