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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. [ABSTRACT FROM AUTHOR]

Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by humans, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real life and problems being often ambiguous, it is very challenging for an expert to exemplify these problems through applying an FG. Vague graph structure (VGS), belonging to the FG family, has good capabilities when facing with problems that cannot be expressed by FGs. VGS can handle the vagueness connected with the incompatible and determinate information of any real-world problem, where FGs may not succeed to bear satisfactory results. The previous definitions' restrictions in FGs have led us to propose new definitions in the VGS. Domination is one of the key issues that has many applications in computer science and social networks. Today, many researchers are trying to prove its application in medical sciences and psychology. Therefore, in this paper, different concepts related to domination in VGSs such as Ni-dominating set, vague full dominating set, minimal N i -DS, and strong capacity Ni-dominating set are defined using some examples. Finally, an application of domination in medical sciences has been presented. [ABSTRACT FROM AUTHOR]

We shall establish some criteria on entire series with finite logarithmic order in terms of maximum term and central index. [ABSTRACT FROM AUTHOR]