A novel application on mutually orthogonal graph squares and graph-orthogonal arrays

El-Mesady, A.; Hamed, Y. S.; Abualnaja, Khadijah M.

doi:10.3934/math.2022410

Cited by 11 publications

(9 citation statements)

References 36 publications

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“…Graph theory is extremely important in complex systems, particularly in computer science. Many scientific areas use graph theory, including engineering, coding [4,5], operational research, biological sciences, and management sciences. For applied scientists and engineers, graph theory is a strong and vital science for evaluating and inventing solutions for a variety of issues.…”

Section: Introductionmentioning

confidence: 99%

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

El-Mesady

Bazighifan

2022

Journal of Function Spaces

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For applied scientists and engineers, graph theory is a strong and vital tool for evaluating and inventing solutions for a variety of issues. Graph theory is extremely important in complex systems, particularly in computer science. Many scientific areas use graph theory, including biological sciences, engineering, coding, and operational research. A strategy for the orthogonal labelling of a bipartite graph G with n edges has been proposed in the literature, yielding cyclic decompositions of balanced complete bipartite graphs K n , n by the graph G . A generalization to circulant-balanced complete multipartite graphs K n , n , ⋯ , n ⏟ m ; m , n ≥ 2 , is our objective here. In this paper, we expand the orthogonal labelling approach used to generate cyclic decompositions for K n , n to a generalized orthogonal labelling approach that may be used for decomposing K n , n , ⋯ , n ⏟ m . We can decompose K n , n , ⋯ , n ⏟ m into distinct graph classes based on the proposed generalized orthogonal labelling approach.

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Section: Introductionmentioning

confidence: 99%

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

El-Mesady

Bazighifan

2022

Journal of Function Spaces

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“…All the constructed results in this paper can be used to generate new graph-orthogonal arrays, new graph-authentication codes, and new graph-transversal designs [3,23]. +ey can also be used in the design of experiments [24].…”

Section: Discussionmentioning

confidence: 99%

“…For standard terminology and notations concerning graph theory, see [1]. Decompositions of complete bipartite graphs have several applications in the design of experiments, graph code generation, and authentication codes [2,3]. Table 1 shows the nomenclature used in the paper.…”

Section: Introductionmentioning

confidence: 99%

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Construction of Mutually Orthogonal Graph Squares Using Novel Product Techniques

El-Mesady

Bazighifan

2022

Journal of Mathematics

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Sets of mutually orthogonal Latin squares prescribe the order in which to apply different treatments in designing an experiment to permit effective statistical analysis of results, they encode the incidence structure of finite geometries, they encapsulate the structure of finite groups and more general algebraic objects known as quasigroups, and they produce optimal density error-correcting codes. This paper gives some new results on mutually orthogonal graph squares. Mutually orthogonal graph squares generalize orthogonal Latin squares interestingly. Mutually orthogonal graph squares are an area of combinatorial design theory that has many applications in optical communications, wireless communications, cryptography, storage system design, algorithm design and analysis, and communication protocols, to mention just a few areas. In this paper, novel product techniques of mutually orthogonal graph squares are considered. Proposed product techniques are the half-starters’ vectors Cartesian product, half-starters’ function product, and tensor product of graphs. It is shown that by taking mutually orthogonal subgraphs of complete bipartite graphs, one can obtain enough mutually orthogonal subgraphs in some larger complete bipartite graphs. Also, we try to find the minimum number of mutually orthogonal subgraphs for certain graphs based on the proposed product techniques. As a direct application to the proposed different product techniques, mutually orthogonal graph squares for disjoint unions of stars are constructed. All the constructed results in this paper can be used to generate new graph-orthogonal arrays and new authentication codes.

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“…Several results of the 𝑂𝐷𝐶s of complete bipartite graphs can be generalized to mutually orthogonal graph squares that have many applications in design theory, graph-orthogonal arrays, and authentication codes [37]. Additionally, the techniques of constructing the 𝑂𝐷𝐶s are considered a tool for graph labeling called orthogonal labeling, which have many applications (for example, see [38]).…”

Section: P Xmentioning

confidence: 99%

On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs

et al. 2023

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Nowadays, graph theory is one of the most exciting fields of mathematics due to the tremendous developments in modern technology, where it is used in many important applications. The orthogonal double cover (ODC) is a branch of graph theory and is considered as a special class of graph decomposition. In this paper, we decompose the complete bipartite graphs Kx,x by caterpillar graphs using the method of ODCs. The article also deals with constructing the ODCs of Kx,x by general symmetric starter vectors of caterpillar graphs such as stars–caterpillar, the disjoint copies of cycles–caterpillars, complete bipartite caterpillar graphs, and the disjoint copies of caterpillar paths. We decompose the complete bipartite graph by the complete bipartite subgraphs and by the disjoint copies of complete bipartite subgraphs using general symmetric starter vectors. The advantage of some of these new results is that they enable us to decompose the giant networks into large groups of small networks with the comprehensive coverage of all parts of the giant network by using the disjoint copies of symmetric starter subgraphs. The use case of applying the described theory for various applications is considered.

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A novel application on mutually orthogonal graph squares and graph-orthogonal arrays

Cited by 11 publications

References 36 publications

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

Decompositions of Circulant-Balanced Complete Multipartite Graphs Based on a Novel Labelling Approach

Construction of Mutually Orthogonal Graph Squares Using Novel Product Techniques

On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs

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