# 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

### Leap Zagreb indices for the Central graph of graph

JPRM-Vol. 17 (2021), Issue 2, pp. 73 – 78 Open Access Full-Text PDF
Ammar Alsinai, Anwar Alwardi, Hanan Ahmed, N. D. Soner
Abstract: The first, second and third leap Zagreb indices are the sum of squares of second degrees of vertices of G, the sum of products of second degrees of pairs of adjacent vertices in G and the sum of products of first and second degrees of vertices of G, respectively. In this Paper We obtained the formal of leap Zagreb Indices for the central graph of graph. Also We compute the the first, second and third leap Zagreb for the central graph of some standard graph.

### Nature of Graphs of Commutative Ring of Gaussian Integer modulo $$n$$ under $$x^{3}-1$$ mapping

JPRM-Vol. 17 (2021), Issue 2, pp. 58 – 72 Open Access Full-Text PDF
Abdul Jalil M. Khalaf, Saima Nazeer, Kishmala Qayyum, Murat Cancan
Abstract: The aim of the present paper is to observe the structures of digraphs derived from the mappings $$f_{1}:Z_{n}[i]\rightarrow Z_{n}[i]$$ defined by $$f_{1}(x)=x^{3}-1$$ whose vertex is $$Z_{n}[i]=\{a+bi:a,b\in Z_{n}\}$$ and for which there is a directed edge from $$x\in Z_{n}[i]$$ to $$y\in Z_{n}[i]$$ if and only if $$x^{3}-1\equiv y\ (mod\ n)$$. In this article, we investigated the structure of digraph. The in-degree of 1 and 0 in $$D_{1}(n)$$ are established where $$D_{1}(n)$$ is digraph obtained. Some regularity conditions of $$D_{1}(n)$$ are also discussed. For certain values of $$n$$, the simple conditions for the number of components and length of cycles is obtained.

### The edge-distance-balanced property of the generalized Petersen graphs $$GP(n,3)$$

JPRM-Vol. 17 (2021), Issue 2, pp. 51 – 57 Open Access Full-Text PDF
Abstract: A graph $$G$$ is said to be edge-distance-balanced if for any edge $$uv$$ of $$G$$, the number of edges closer to $$u$$ than to $$v$$ is equal to the number of edges closer to $$v$$ than to $$u$$. It is proven that for any integers $$j\geq2$$, the generalized Petersen graphs $$GP(6j+9,3),$$ $$GP(6j+10,3),$$ $$GP(6j+11,3),$$ $$GP(6j+12,3)$$ and $$GP(6j+13,3)$$ are not edge-distance-balanced. The aim of this paper is to give some interesting facts about the edge-distance-balanced property of the generalized Petersen graphs $$GP(n,3)$$.

### Distance and Degree Based Topological Polynomial and Indices of X-Level Wheel Graph

JPRM-Vol. 17 (2021), Issue 2, pp. 39 – 50 Open Access Full-Text PDF
Ali Hasan, Muhammad Hakim Ali Qasmi, Ammar Alsinai, Mehdi Alaeiyan, Mohammed Reza Farahani, Murat Cancan
Abstract: In this paper we discussed the partitioning of the wheel graph and we calculate the M-polynomial, Hosoya polynomial, Harary polynomial, Schultz polynomial, Modified Schultz polynomial, Eccentric connectivity polynomial, Modified Wiener index, Modified Hyper Wiener index, Generalized Harary index, Multiplicative Wiener index, Schultz index, Modified Schultz index, Eccentric connectivity index and also derived the Randic index, Generalized Randic index, First Zagreb, Second Zagreb, Second Modified Zagreb, General Randic and Inverse General Randic, Harmonic, Symmetric Division and Inverse Sum index of generalized wheel networks $$\mathbb W_{x,y}$$.

### Mathematical Modeling of Transmission of Water Pollution

JPRM-Vol. 17 (2021), Issue 2, pp. 20 – 38 Open Access Full-Text PDF
Ebenezer Bonyah, Paul Agbekpornu, Canan Unlu
Abstract: Water pollution is one of the major environmental problems facing developing countries all over the planet. The mathematical model for soluble and insoluble water pollutants has been formulated in light of ordinary differential equations. Sensitivity analysis is carried out to examine parameter values that are associated with the reproduction number in order to minimize water pollution. The basic reproduction number of the water pollution model is determined and the steady state of the model is investigated. Numerical analysis indicates that water treatment is an essential tool for obtaining quality water for human consumption.

### The Sub(G)-Graph of Direct Product two Dihedral Groups

JPRM-Vol. 17 (2021), Issue 2, pp. 15 – 19 Open Access Full-Text PDF
Ola Hasan Al- Lahaibat, Mohsin ShalanAbdulhussein, Ahmed M. AL-obaid, Hayder Baqer Ammen
Abstract: In this paper, we introduce a new graph, called subgroup product graph $$Sub(G)$$ of a group $$G$$. The set of vertices of $$Sub(G)$$ is the set of all subgroups of $$G$$ $$V(Sub(G))=\{H_i \mid H_i \leq G , \forall i\}$$ and two vertices $$H_i$$ and $$H_j$$ are adjacent if and only if $$H_i \cap H_j \neq \emptyset$$. We characterize when $$Sub(G)$$ is connected and complete graph.