# Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing two issues in June and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

### 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.

### Vertex equitable labeling for ladder and snake related graphs

JPRM-Vol. 1 (2017), Issue 1, pp. 01 – 07 Open Access Full-Text PDF
A. Lourdusamy, F. Patrick
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.