## Abstract

Balaban index is defined as *G*, *n* and *m* are the cardinalities of the vertex and the edge set of *G*, respectively, and *w*(*u*) (resp. *w*(*v*)) denotes the sum of distances from *u* (resp. *v*) to all the other vertices of *G*. In 2011, H. Deng found an interesting property that Balaban index is a convex function in double stars. We show that this holds surprisingly to general graphs by proving that attaching leaves at two vertices in a graph yields a new convexity property of Balaban index. We demonstrate this property by finding, for each *n*, seven trees with the maximum value of Balaban index, and we conclude the paper with an interesting conjecture.

## Abstract

Let *G* be a graph and let *m _{ij}*(

*G*)

*, i, j ≥*1, represents the number of edge of

*G*, where

*i*and

*j*are the degrees of vertices

*u*and

*v*respectively. In this article, we will compute different polynomials of flower graph

*f*(

*), namely*

_{n×m}*M polynomial*and

*Forgotten*polynomial. These polynomials will help us to find many degree based topological indices, included different Zagreb indices, harmonic indices and forgotten index.

## Abstract

In this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

## Abstract

There is no mathematical solution to adding up transcendental functions other than numerical process. This paper put forward analytical method to model the sum of sigmoid like functions with an equivalent function. The *Brillouin* and *Langevin* as well as the *error* function, the *tanh*, *sigmoid* and the *tan ^{-}*

^{1}functions are investigated, their equivalent functions are calculated for four components and the error between the numerical (computer assisted) result and the equivalent function is tested for accuracy. The best modelling function, the most useful to include into mathematical operations, is pointed out finally, based on its performance and convenience. The paper intends to help people involved mostly in modelling hysteresis in Magnetism and other field of research in physics.

## Abstract

In this manuscript, we shall apply the tools and methods from optimal control to analyze various minimally parameterized models that describe the dynamics of populations of cancer cells and elements of the tumor microenvironment under different anticancer therapies. In spite of their simplicity, the analysis of these models that capture the essence of the underlying biology sheds light on more general scenarios and, in many cases, leads to conclusions that confirm experimental studies and clinical data. We focus on four applications: optimal control applied to compartmental models, brain tumors, drug resistance and antiangiogenic treatment.

## Abstract

This paper presents an analytical method to determine the rise-set times of satellite-satellite visibility periods in different orbits. The Visibility function in terms of the orbital elements of the two satellites versus the time were derived explicitly up to *e*
^{4}. The line-of-sight corrected for Earth Oblateness up to *J*
_{2}, were considered as a perturbation to the orbital elements. The visibility intervals of the satellites were calculated for some numerical examples in order to test the results of the analytical work.

## Abstract

In this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions *K*
_{5} and *K*
_{12} by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

## Abstract

In a previous (herein referred to as Ammar, Amin and Hassan Paper [1]) the statement of the problem was formulated and the basic visibility function between two satellites in terms of the orbital elements and time were derived. In this paper the perturbing effect due to drag force on the visibility function were derived explicitly up to *O*(*e*
^{4}), by using Taylor’s expansion for the visibility function about certain epoch. We determine the rise and set times of the satellites through the sign of the visibility function. Numerical examples were worked out for some satellites in order to check the validity of the work.

## Abstract

In this paper we introduce the walk polynomial to find the number of walks of different lengths in a simple connected graph. We also give the walk polynomial of the bipartite, star, wheel, and gear graphs in closed forms.

## Abstract

In this article, we have considered for numerical solution of a Poisson and Laplace equation in a domain. we have presented a novel finite difference method for solving the system of the boundary value problems subject to Dirichlet boundary conditions. We have derived the solution of the Poisson and Laplace equations in a two-dimensional finite region. We present numerical experiments to demonstrate the efficiency of the method.