###### Abstract  
Recent developments in commutative algebra, linear algebra, and graph theory allow us to approach various issues in several fields. Circulant graphs now have a wider range of practical uses, including as the foundation for optical networks, discrete cellular neural networks, small-world networks, models of chemical reactions, supercomputing and multiprocessor systems. Herein, we are concerned with the decompositions of the bipartite circulant graphs. We propose the Cartesian and tensor product approaches as helping tools for the decompositions. The proposed approaches enable us to decompose the bipartite circulant graphs into many categories of graphs. We consider the use cases of applying the described theory of bipartite circulant graph decomposition to the problems of finding new topologies and deadlock-free routing in them when building supercomputers and networks-on-chip. addressing, missile guidance, and other complex problems that are difficult to describe in today's reality. Chemical graph theory, a fascinating aspect of mathematics, is a brand-new area of modern research. The chemical molecules are mathematically described as a molecular graph. In a molecular graph, vertices stand for atoms and edges for chemical bonds. The topological and structural characteristics of these molecular structures are investigated using several techniques of graph theory. In order to mathematically forecast the persuasive benefits of the associated chemical molecule, the topology of the molecular structure is necessary [17]. One of the most recent research areas among scientists is the study of chemical compounds using mathematical modeling [18; 19]. Therefore, using mathematical techniques to investigate chemical compounds, such as combinatorics and topology, is crucial in practical research. In order to suit the needs of chemists, many topological descriptors are introduced nowadays [20; 21]. Let us introduce some notations. Edge set \(E\) and vertex set \(V\) are the two sets that make up the graph \(G=(V,\ E)\). If there are only finitely many elements in both of these sets, then \(G\) is the finite graph. Otherwise, it is an infinite graph. A simple graph is the one that has no multiple edges and loops, and it is undirected if no directions are indicated by the edges. If the adjacency matrix of a simple finite graph is circulant, researchers can refer to the graph as being circulant. For instance, Mobius ladders and Paley graphs are circulant graphs (Figure 1). Numerous applications, such as locating multiprocessor faults, intruding in buildings and facilities, as well as environmental monitoring employing wireless sensor networks, have drawn attention to location detection issues. In each of these scenarios, the system or structure can be represented as a graph. Parallel networks are frequently modeled using circulant graphs. Circulant graphs class is one of the most important classes of graphs [22; 23; 24; 25]. Over the past few decades, circulant graphs have received a lot of attention. The class of circulant graphs includes complete graphs and traditional ring topologies. There are a lot of papers handled the circulant graphs' algebraic characteristics. Circulant graphs have been proposed for numerous network applications, including local area computer networks, parallel processing architectures, VLSI design, and distributed computing, from a more practical standpoint. Some traditional distributed and parallel systems are built on circulant graphs of varying degrees [26; 27]. Circulant networks' range of useful applications has recently expanded. They are now used as the structural foundation for models of chemical  
Figure 1: The Paley graph of order 13.  
reactions [28], small-world network models [29], multi-processor cluster systems [30], optical networks [31], discrete cellular neural networks [32], and other models. The current study fields still revolve around the characterization, analysis, and applications of circulant graphs.