The latest developments in algebra and graph theory allow us to ask a natural question, what is the application in real world of this graph associated with some mathematical system? Groups can be used to construct new non-associative algebraic structures, loops. Graph theory plays an important role in various fields through edge labeling. In this paper, we shall discuss some applications of bipartite graphs, related with Latin squares of Wilson loops, such as metabolic pathways, chemical reaction networks, routing and wavelength assignment problem, missile guidance, astronomy and x-ray crystallography.

Lot of achievements have been done to comprehend the latest phenomena of our life by discussing the interrelationship of two boughs, graph theory and algebra, of mathematics. In the modern time, it seems to be of great importance to study about eminent, extensive and certifiable applications through well known tools of graph theory. Combinatorial commutative algebra, moderately new, quickly creating numerical tool and foremost tendril of mathematics, has been used exhaustively to solve a list of open problems in algebra under the shadow of graphs. It can be helpful, instead of algebraically, to find the metric dimension of zero-divisor graphs, NP-complete problem, related with finite mathematical structures. In this article we shall discuss some graphs, zero-divisor graphs of finite Boolean rings and balanced bipartite graphs of conjugacy closed loops, and their associated applications.

Taking into account the most recent improvements in graph theory and algebra, we can associate graphs of some mathematical structures with certifiable, widely known applications. This paper seeks to explore the connections established through edge labeling among Latin squares derived from Moufang quasigroups, which are constructed using additive abelian and multiplicative groups, along with their substructures and complete bipartite graphs. The algebraic characteristics of quasigroups exhibiting the antiautomorphic inverse property have been extensively examined in this study. These characteristics encompass identities associated with fixed element maps. To analyze the behavior of these groups under holomorphism, we utilize similar conditions.

In mathematical chemistry, the algebraic polynomial serves as essential for calculating the most accurate expressions of distance-based, degree-distance-based, and degree-based topological indices. The chemical reactivity of molecules, which includes their tendency to engage in particular chemical processes or go through particular reactions, can be predicted using topological indices. Considerable effort has been put into examining the many topological descriptors of simple graphs using ring structures and well-known groups instead of nonassociative algebras, quasigroups, and loops. Both finite quasigroups and loops are the generalizations of groups. In this article, we calculate topological descriptors and some well-known polynomials, M -polynomial, Hosoya’s polynomial, Schultz’s polynomial, and modified Schultz polynomial of finite relatively prime graphs of most orders connected with two classes of quasigroups and go through their graphical aspects.