The analysis of networks and graphs through topological descriptors carries out a useful role to derive their underlying topologies. This process has been widely used in biomedicine, cheminformatics, and bioinformatics, where assessments based on graph invariants have been made available for effectively communicating with the various challenging schemes. In the studies of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs), graph invariants are used to approximate the biological activities and properties of chemical compounds. In this paper, we give the results related to the eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, eccentric adjacency index, modified eccentric-connectivity index, eccentric distance sum, Wiener index, Harary index, hyper-Wiener index and degree distance index of a new graph operation named as “subdivision vertex-edge join” of three graphs.

Let γi(C_m ⎕ C_n) denote the independent domination number of the strong product of the graphs C_m, the cycle of length m, m ≥ 3 and the graph C_n, the cycle of length n, n ≥ 3. In this paper, we determine the exact values of the independent domination numbers of C_m ⎕ C_n with m ≤ n when 3 ≤ m ≤ 8 and n arbitrary large.

The present study explores the characteristics of 2D MHD melting with reference to mass and heat transportation upon stagnation point Powell–Eyring nanofluid flow on an extensible surface. Melting is an important phenomenon that is involved in many procedures such as permafrost melting, solidification of slag, defrosting frozen ground etc., all of which are examples of soil freezing and melting that involve heat trafficking through a coil in a grounded pump. A mathematical model is developed for the boundary layer flow. The differential equations are solved through a numerical algorithm which makes use of the boundary value problem solver bvp4c, applying MATLAB software. The numerical variations of embedded parameters on velocity lineation, temperature figuration, and concentration delineation are represented graphically, as are the width of the boundary layer value and the delineation rate for the increasing velocity parameter. The velocity function shows a decremental response for M while the opposite behavior is seen against the concentration field.

Graph theory plays an important role in modeling and designing any chemical structure or network. The molecular topological indices are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper we have computed vertex-Szeged index, vertex PI index,the vertex Szeged geometricarithmetic index and the weighted vertex Szeged indexfor the crystal cubic structure CCC(n) of carbon.

In this paper, certain Hermite–Hadamard–Mercer-type inequalities are proved via Riemann–-Liouville fractional integral operators. We established several new variants of Hermite–Hadamard’s inequalities for Riemann–Liouville fractional integral operators by utilizing Jensen–Mercer inequality for differentiable mappingϒwhose derivatives in the absolute values are convex. Moreover, we construct new lemmas for differentiable functionsϒ′,ϒ″, andϒ‴and formulate related inequalities for these differentiable functions using variants of Hölder’s inequality.

The representation of chemical compounds and chemical networks with the M-polynomials is a new idea, and it gives nice and good results of the topological indices. These results are used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this article, particular attention will be put on the derivation of M-polynomial for the benzene ring embedded in the P-type surface network in 2D. Furthermore, the topological indices based on the degrees are also derived by using the general form of M-polynomial of the benzene ring embedded in the P-type surface network BRm,n. In the end, the graphical representation and comparison of the M-polynomial and the topological indices of the benzene ring embedded in the P-type surface network in 2D are described.