Abstract
During the last two centuries, after the question asked by Euler concerning mutually orthogonal Latin squares (MOLS), essential advances have been made. MOLS are considered as a construction tool for orthogonal arrays. Although Latin squares have numerous helpful properties, for some factual applications these structures are excessively prohibitive. The more general concepts of graph squares and mutually orthogonal graph squares (MOGS) offer more flexibility. MOGS generalize MOLS in an interesting way. As such, the topic is attractive. Orthogonal arrays are essential in statistics and are related to finite fields, geometry, combinatorics and error-correcting codes. Furthermore, they are used in cryptography and computer science. In this paper, our current efforts have concentrated on the definition of the graph-orthogonal arrays and on proving that if there are k MOGS of order n, then there is a graph-orthogonal array, and we denote this array by G-OA(n2,k,n,2). In addition, several new results for the orthogonal arrays obtained from the MOGS are given. Furthermore, we introduce a recursive construction method for constructing the graph-orthogonal arrays.
Highlights
A graph is a couple G = (U, E), where U is a set of vertices and E is a set of edges, and E ⊆ U × U.The two ends of an edge are called two adjacent vertices
Orthogonal arrays are essential in statistics where they are basically utilized in experimental design, they are immensely important in medicine, manufacturing and agriculture
mutually orthogonal graph squares (MOGS) are considered a generalization of the mutually orthogonal Latin squares (MOLS)
Summary
Latin square with order is an × matrix whose entries are taken from a set with elements of A appear time inone eachtime row in and each column. Let us indicate the entry at row i and Hereafter, we will need the Kronecker product of the graph squares. An N × k matrix A whose entries are taken from S is called an orthogonal array with s levels, strength t and index λ (for some t in the range 0 ≤ t ≤ k) if all N × t subarrays of A containing each t-tuple rely on S precisely λ times in a row.
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