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The application of graph theory in the study of molecular physical and chemical properties involves theoretical mathematical chemistry. Atoms, represented by vertices, and edges, represented by bonds between them, are detailed in simple graphs called chemical graphs. The mathematical derivation of the numerical value of a graph is called the molecular descriptor of the graph. Any connected graph wherein no edge is contained in exclusive of a single cycle is called a cactus graph. In the research in this article, expressions for various molecular descriptors of line graph of the graph obtained by the rooted product of the cycle and path graphs are constructed. This article obtained the calculation of molecular descriptors for line graphs of chain ortho cactus and chain para cactus graphs. To predict the biological activity of a compound, the generalized Zagreb index, the first Zagreb index M1G, the second Zagreb index M2G, the F-index, the general Randic index, the symmetric division, the atom bond connectivity (ABC), and the geometric arithmetic (GA) descriptors are created.

In view of the successful applications of graph theory, relationships between the biological activity and chemical structure have been developed. One of the popular topics in graph theory is problems relating to topological indices. Degree-based topological indices, distance-based topological indices, and counting-related topological indices are various types of topological indices. Physiochemical properties such as boiling point and stability of chemical compounds are correlated by these topological indices. A topological index of a graph is a numerical quantity obtained from the graph mathematically. A cactus graph is a connected graph in which no edge lies in more than one cycle. In this study, we have derived certain degree-based topological indices for some families of graphs consisting of graph obtained by the rooted product of paths and cycles and two types of cactus graph (paracactus and orthocactus) with the help of the generalized Zagreb index.

In this paper, the velocity field and the time dependent shear stress of Maxwell fluid with Caputo fractional derivatives in an infinite long circular cylinder of radius R is discussed. The motion in the fluid is produced by the circular cylinder. The fluid is initially at rest and at time t=0+, cylinder begins to oscillate with the velocity fsinωt, due to time dependent shear stress acting on the cylinder tangentially. The hybrid technique used in this paper for the solution of the problem has less computational efforts and time cost as compared to other commonly used methods. The obtained solutions are in transformed domain, which are expressed in terms of modified Bessel functions I0(·) and I1(·). The inverse Laplace transformation has been calculated numerically by using MATLAB package. The semi analytical solutions for Maxwell fluid with fractional derivatives are reduced to the similar solutions for Newtonian and ordinary Maxwell fluids as limiting cases. In the end, numerical simulations have been performed to analyze the behavior of fractional parameter α, kinematic viscosity ν, relaxation time λ, radius of the circular cylinder R and dynamic viscosity μ on our obtained solutions of velocity field and shear stress. Keywords: Shear stress, Maxwell fluid, Laplace transformation, Velocity field, Modified Bessel function