### On some parameters related to fixing sets in graphs

JPRM-Vol. 1 (2018), Issue 1, pp. 01 – 12 Open Access Full-Text PDF
Abstract: The fixing number of a graph G is the smallest cardinality of a set of vertices $$F ⊆ V (G)$$ such that only the trivial automorphism of $$G$$ fixes every vertex in $$F$$. In this paper, we introduce and study three new fixing parameters: fixing share, fixing polynomial and fixing value.

JPRM-Vol. 1 (2017), Issue 1, pp. 90 – 106 Open Access Full-Text PDF
Syeda Laila Naqvi, Jeremy Levesley, Salma Ali
Abstract: We propose a meshless adaptive solution of the time-dependent partial differential equations (PDE) using radial basis functions (RBFs). The approximate solution to the PDE is obtained using multiquadrics (MQ). We choose MQ because of its exponential convergence for sufficiently smooth functions. The solution of partial differential equations arising in science and engineering frequently have large variations occurring over small portion of the physical domain. The challenge then is to resolve the solution behaviour there. For the sake of efficiency we require a finer grid in those parts of the physical domain whereas a much coarser grid can be used otherwise. Local scattered data reconstruction is used to compute an error indicator to decide where nodes should be placed. We use polyharmonic spline approximation in this step. The performance of the method is shown for numerical examples of one dimensional Kortwegde-Vries equation, Burger’s equation and Allen-Cahn equation.

### Analysis of steady non isothermal two dimensional flow of second grade fluid in a constricted artery

JPRM-Vol. 1 (2017), Issue 1, pp. 75 – 89 Open Access Full-Text PDF
A.A. Mirza, A.M. Siddiqui, T. Haroon
Abstract: Steady analytical solution of non-isothermal, second grade fluid through an artery having constriction of cosine shape in two dimension is presented. The governing equations are transformed into stream function formulation which are solved analytically with the help of regular perturbation technique. The solutions thus obtained are presented graphically in terms of streamlines, wall shear stress, separation points, pressure gradient and temperature distribution. It is observed that an increase in height of constriction $$(\in)$$ gives rise in wall shear stress, pressure gradient and temperature, whereas critical Reynolds  number $$(R_e)$$ decreases. Further an increase in second grade parameter $$(α)$$ increases the temperature, pressure gradient, velocity and wall shear stress while critical Re decreases. Its worthy to mention that the present results are compared with the already published results which ensures good agreement.

### Exact solutions of time fractional free convection flows of viscous fluid over an isothermal vertical plate with caputo and caputo-fabrizio derivatives

JPRM-Vol. 1 (2017), Issue 1, pp. 56 – 74 Open Access Full-Text PDF
Nehad Ali Shah, M. A. Imran, Fizza Miraj
Abstract: The unsteady time fractional free convection flow of an incompressible Newtonian fluid over an infinite vertical plate due to an impulsive motion of the plate and constant temperature at the boundary is analyzed. The old (Caputo) and new (Caputo-Fabrizio) fractional derivative approaches have been used to develop a physical model and a comparison has been drawn between their solutions. Boundary layers equations in non dimensional form are solved analytically by the Laplace transform technique. Exact solutions for velocity and temperature are obtained in terms of Wrights function. The expressions for rate of heat transfer in both cases are also determined. Solutions for integer order derivatives are obtained as limiting case. Numerical computations were made through software Mathcad and observed some physical aspects of fractional and material parameters are presented. It is found that the rate of heat transfer of Caputo Fabrizio model have higher values than Caputo one as we increased the value of fractional parameter and fractional fluids tend to superpose to that of ordinary fluid.

### A ninth-order iterative method for nonlinear equations along with polynomiography

JPRM-Vol. 1 (2017), Issue 1, pp. 41 – 55 Open Access Full-Text PDF
Waqas Nazeer, Abdul Rauf Nizami, Muhammad Tanveer, Irum Sarfraz
Abstract: In this paper, we suggest a new ninth order predictor-corrector iterative method to solve nonlinear equations. It is also shown that this new iterative method has convergence of order nine and has efficiency index 1.7321. Moreover, some examples are given to check its validity and efficiency. Finally, we present polynomiographs for some complex polynomials via our new method.

### Computation of hosaya polynomial, wiener and hyper wiener index of jahangir graph $$j_{6,m}$$

JPRM-Vol. 1 (2017), Issue 1, pp. 30 – 40 Open Access Full-Text PDF
Mehdi Rezaei, Mohammad Reza Farahani, Waqas Khalid, Abdul Qudair Baig
Abstract: Let $$G = (V, E)$$ be a simple connected graph with vertex set $$V$$ and edge set $$E$$. For two vertices $$u$$ and $$v$$ in a graph $$G$$, the distance $$d(u, v)$$ is the shortest path between $$u$$ and $$v$$ in $$G$$. Graph theory has much advancements in the field of theoretical chemistry. Recently, chemical graph theory is becoming very popular among researchers because of its wide applications of mathematics in chemistry. One of the important distance based topological index is the Wiener index, defined as the sum of distances between all pairs of vertices of $$G$$, defined as $$W(G) = \sum_{ u,v∈V (G)} d(u, v)$$. The Hosaya polynomial is defined as $$H(G, x) =\sum _{u,v∈V (G)} x ^{d(u,v)}$$. The hyper Wiener index is defined as $$WW(G) =\sum_{u,v∈V (G)} d(u, v) + \frac{1}{2}\sum_{u,v∈V (G)}d^{2}(u, v)$$. In this paper, we study and compute Hosaya polynomial, Wiener index and hyper Wiener index for Jahangir graph $$J_{6,m}$$, $$m ≥ 3$$. Furthermore, we give exact values of these topological indices.

### Properties of co-intersection graph of submodules of a module

JPRM-Vol. 1 (2017), Issue 1, pp. 16 – 29 Open Access Full-Text PDF
Lotf Ali Mahdavi, Yahya Talebi
Abstract: Let $$R$$ be a ring with identity and $$M$$ be a unitary left Rmodule. The co-intersection graph of proper submodules of $$M, Ω(M)$$ is an undirected simple graph whose vertices are non-trivial submodule of $$M$$ in which two vertices N and K are joined by an edge, if and only if $$N + K \neq M$$. In this paper, we study several properties of $$Ω(M)$$. We prove that, if $$Ω(M)$$ is a path, then $$Ω(M) \cong P_2$$or $$Ω(M) \cong P_3$$. We show that, if $$Ω(M)$$ is a forest, then each component of $$Ω(M)$$ is complete or star graph. We determine the conditions under which $$Ω(M)$$ is weakly perfect. Moreover, we introduce the universal vertices and the dominating sets of $$Ω(M)$$ and their relationship with the non-trivial small submodules of $$M$$.

### Narumi-Katayama and modified Narumi-Katayama indices of graphs

JPRM-Vol. 1 (2017), Issue 1, pp. 08 – 15 Open Access Full-Text PDF
Abstract: Let $$G$$ be a simple connected molecular graph in chemical graph theory, then its vertices correspond to the atoms and the edges to the bonds. Chemical graph theory is an important branch of graph theory, such that there exits many topological indices in it. Also, computing topological indices of molecular graphs is an important branch of chemical graph theory. Topological indices are numerical parameters of a molecular graph $$G$$ which characterize its topology. In the present study we compute and report several results of the Narumi-Katayama and modified NarumiKatayama indices for some widely used chemical molecular structures.
Abstract: Let $$G$$ be a graph with p vertices and q edges and $$A = {0, 1, 2, · · · ,\frac{q}{2}}$$. A vertex labeling $$f : V (G) → A$$ induces an edge labeling $$f^∗$$ defined by $$f^∗ (uv) = f(u) + f(v)$$ for all edges $$uv$$. For $$a ∈ A$$, let $$v_f (a)$$ be the number of vertices $$v$$ with$$f(v) = a$$. A graph $$G$$ is said to be vertex equitable if there exists a vertex labeling f such that for all $$a$$ and $$b$$ in A, $$|v_f (a) − v_f (b)| ≤ 1$$ and the induced edge labels are $${1, 2, 3, · · · , q}$$. In this paper, we prove that triangular ladder $$T L_n, L_n ⊙ mK_1, Q_n ⊙ K_1, T L_n⊙K_1$$ and alternate triangular snake $$A(T_n)$$ are vertex equitable graphs.