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Let m and n be two positive integers. We use [m] for the set { 1 , ... , m } . For any integer i, i m designates the minimum nonnegative integer that is congruent to i modulo m, and S m , i n stands for the set of those x ∈ [ m ] [ n ] satisfying ∑ t = 1 n x (t) is congruent to i modulo m. An enumeration of S m , i n as x 1 , ... , x m n - 1 is called a tight balanced cyclic sequence if ∑ p ∈ [ m n - 1 ] x p + 1 (t) - x p (t) m = m n n holds for each t ∈ [ n ] , where the subscript m n - 1 + 1 should be understood as 1. Assuming that m n is a multiple of n, is there always a tight balanced cyclic sequence over S m , i n ? We provide a positive answer to this question when n is a power of m. [ABSTRACT FROM AUTHOR]

The concept of detecting arrays was developed to locate and detect interaction faults arising between the factors in a component-based system during software testing. In this paper, we propose a family of consecutive detecting arrays (CDAs) in which the interactions between factors are considered to be ordered. CDAs can be used to generate test suites for locating and detecting interaction faults between neighboring factors. We establish a general criterion for measuring the optimality of CDAs in terms of their size. Based on this optimality criterion, the equivalence between optimum CDAs and consecutive orthogonal arrays with prescribed properties is explored. Using the advantages of this equivalence, a great number of optimum CDAs are presented. In particular, the existence of optimum CDAs with few factors is completely determined. [ABSTRACT FROM AUTHOR]

One-overlapped factorizations of finite groups were introduced by Shinohara (Linear Algebra Appl. 436(4):850-857, 2012) for constructing a new class of thin Lehman matrices. However, all known examples and constructions arise from cyclic or dihedral groups. In this paper, we prove the existence of one-overlapped factorizations of non-abelian groups that are not dihedral and describe a simple construction. The corresponding incidence structures and Lehman matrices are also considered. [ABSTRACT FROM AUTHOR]

We define the type of graph products, which enable us to treat many graph products in a unified manner. These unified graph products are shown to be compatible with Godsil-McKay switching. Furthermore, by this compatibility, we show that the Doob graphs can also be obtained from the Hamming graphs by switching. [ABSTRACT FROM AUTHOR]

Let $$M=(m_{i,j})$$ be a magic square, where $$0\le m_{i,j}\le n^2-1$$ , $$0\le i, j\le n-1$$ . M is called diagonally ordered if both the main diagonal and the back diagonal, when traversed from left to right, have strictly increasing values. Let $$M=nA+B$$ , where $$A=(a_{i,j})$$ , $$B=(b_{i,j})$$ , $$0\le a_{i,j}, \ b_{i,j}\le n-1$$ for $$0\le i, j\le n-1$$ . M is called rational if both A and B possess the property that the sums of the n numbers in every row and every column are the same; otherwise, M is said to be irrational. In this paper, a pair of weakly diagonally ordered irrational orthogonal matrices (WDOIOM for short) is introduced to construct an irrational diagonally ordered magic square (IDOMS). It is proved that there exists a WDOIOM( n) for each positive integer $$n\ge 5$$ , and there does not exist a WDOIOM( n) for $$n\in \{2,3,4\}$$ . Consequently, it is proved that there exists an IDOMS( n) for each positive integer $$n\ge 5$$ , and there does not exist an IDOMS( n) for $$n\in \{2,3,4\}$$ . [ABSTRACT FROM AUTHOR]

We consider the problem of completing a $$(0,-1)$$ -matrix to an alternating sign matrix (ASM) by replacing some $$0$$ s with $$-1$$ s. An algorithm can be given to determine a completion or show that one does not exist. We are concerned primarily with bordered-permutation $$(0,-1)$$ matrices, defined to be $$n\times n$$ $$(0,-1)$$ -matrices with only $$0$$ s in their first and last rows and columns where the $$-1$$ s form an $$(n-2)\times (n-2)$$ permutation matrix. We show that any such matrix can be completed to an ASM and characterize those that have a unique completion. [ABSTRACT FROM AUTHOR]