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Graph theory has provided a very useful tool, called topological index, which is a number from the graph M with the property that every graph N isomorphic to M value of a topological index must be same for both M and N. Topological index is a descriptor in graph theory which is used to quantify the physio-chemical properties of the chemical graph. In this paper, we computed closed forms of M-polynomials for line graphs of H-naphtalenic nanotubes and chain silicate network. From M-polynomial, we obtained some topological indices based on degrees.

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In this paper, we consider conservation laws and exact solutions of the (3+1)-dimensional modified KdV–Zakharov–Kuznetsov equation. Firstly, we construct conservation laws of the given equation with the help of the conservation theorem; the developed conservation laws are modified conservation laws. Then, we obtain exact solutions of the given equation via the (G′/G,1/G)-expansion method. The obtained solutions are classified as trigonometric solutions, hyperbolic solutions and rational solutions. Furthermore, graphical representations of the obtained solutions are given.

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<abstract><p>In the present paper, a Lie-group integrator, based on $ GL(4, \mathbb{R}) $ has been newly constructed to consider the flow characteristics in an electrically conducting second grade fluid over a stretching sheet. Present method which have a very fast convergence, permits us to explore some missing initial values at the left-end. Accurate initial values can be achieved when the determined target equation is valid, and then we can apply the group preserving scheme (GPS) as a geometric approach to obtain a rather accurate numerical solution. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation point flow and stretching of the sheet parameters are illustrated.</p></abstract>