Almost all schemes existed in the literature to solve the Lattice Boltzmann Equation like stream & collide, finite difference, finite element, finite volume schemes are explicit. However, it is known fact that implicit methods utilizes better stability and faster convergence compared to the explicit methods. In this paper, a method named herein as Implicit Finite Volume Lattice Boltzmann Method (IFVLBM) for incompressible laminar and turbulent flows is proposed and it is applied to some 2D benchmark test cases given in the literature. Alternating Direction Implicit, an approximate factorization method is used to solve the obtained algebraic system. The proposed method presents a very good agreement for all the validation cases with the literature data. The proposed method shows good stability characteristics, the CFL number is eased. IFVLBM has about 2 times faster convergence rate compared with Implicit-Explicit Runge Kutta method even though it possesses a computational burden from the solution of algebraic systems of equations.

In this study, we compute Merrifield-Simmons index of graphs which corresponds to PL (2, n) by using the decomposition formula. Examination shows that Merri-field-Simmons index of certain classes of graphs deduce as a result in a difference equation. Further, the prominent formula for Merrifield-Simmons index of PL (2, n) and generating function are found as a function of the number n of hexagons in the state of PL (2, n). Also the relations formed for the (PL (2, n, n)) graphs obtained by combining two PL (2, n) with a certain angle between them are given.

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