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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. [ABSTRACT FROM AUTHOR]

In this paper, we propose an innovative approach to determine the approximate solution of the coupled time-fractional Keller-Segel (K-S) model. We use the fractional complex transform (FCT) to switch the model into its differential partner, and then, the homotopy perturbation method (HPM) is introduced to tackle its nonlinear elements using He's polynomials. This two-scale theory helps to define the physical meaning of the FCT for the solution of the K-S model. Some examples are illustrated to show that the proposed scheme presents the significant results. The considerable findings show that this strategy does not require any assumptions and also reduces the massive computations without imposing any constraints. This technique is also suitable in functional studies of fractal calculus due to its powerful and robust support for nonlinear problems. [ABSTRACT FROM AUTHOR]