# 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

### A new parallel numerical algorithm for solving singular Perturbation problems in partial differential equations

JPRM-Vol. 17 (2021), Issue 2, pp. 111 – 122 Open Access Full-Text PDF
Khalid Mindeel Mohammed Al-Abrahemee, Madeha Shaltagh Yousif
Abstract: In this study, a new method based on a neural network has been used as a solution for singular perturbation problems in partial differential equations (SPPDEs). Specifically, a modified neural network with a parallel numerical algorithm was used to train the Levenberg-Marquardt (L-M) algorithm with new data and hypotheses. This method is generally applicable to SPPDEs. We consider the convergence under $$\vartheta_{k}=\min (\|E_{k}\|,\|J_{k}^{T}E_{k}\|)$$ of the L-M algorithm, where $$\|E_{k}\|$$ provides a local error bound, and $$J(\varpi _{k})=E^{‘}(\varpi _{k})$$ is the Jacobian. The sequence generated by the L-M algorithm converges to the solution set quadratically. We use some examples to prove that the proposed algorithm, when implemented in MATHEMATICA 11.2, is more efficient and accurate than the standard algorithm.

### Reciprocal leap indices of some wheel related graphs

JPRM-Vol. 17 (2021), Issue 2, pp. 101 – 110 Open Access Full-Text PDF
Swamy Javaraju, Ammar Alsinai, Anwar Alwardi, Hanan Ahmed, N. D. Soner
Abstract: Recently, Ammar Alsinai et al., [1], introduced Reciprocal leap Zagreb indices of a graph based on the inverse second degree of vertices. The first Reciprocal leap Zagreb index $$RL_{1}(G)$$ is equal to the sum of squares of the inverse second degrees of the vertices, the second Reciprocal leap Zagrab index $$RL_{2}(G)$$ is equal to the sum of the products of the inverse second degrees of pairs of adjacent vertices of $$G$$ and the third Reciprocal leap Zagreb $$RL_{3}$$ is equal to the sum of the products of the inverse first degrees with the inverse second degrees of the vertices. In this paper, exact expression for Reciprocal leap Zagreb indices of wheel $$w_{n}$$, and some related graphs as gear $$G_{n}$$, helm $$H_{n}$$, flower $$fl_{n}$$ and sunflower $$sf_{n}$$ graphs are commuted.

### DG-domination topology in Digraph

JPRM-Vol. 17 (2021), Issue 2, pp. 93 – 100 Open Access Full-Text PDF
Khalid Sh. Al’Dzhabri, Ahmed A. Omran, Manal N. Al-Harere
Abstract: Throughout this paper, a new entry from domination approaches is introduced which is called a $$DG-$$ dominating set and which builds its dominance by the topology that relates to digraphs called $$\tau _{DG}-$$ Topological space. Also, a modern definition of domination number called $$DG-$$domination number is created. Moreover, some properties of $$DG-$$ dominating set are presented. Finally, the $$DG-$$domination number for certain graphs are determined.

### Geometric arithmetic index of Alkanes and Unicyclic chemical graphs

JPRM-Vol. 17 (2021), Issue 2, pp. 85 – 92 Open Access Full-Text PDF
Mohanad A. Mohammed, Ahmed Abdulhussein Jabbar
Abstract: The geometric arithmetic $$GA$$ index is one of the most investigated degree based molecular structure descriptors that have applications in chemistry. For a graph $$G$$, the geometric arithmetic $$GA$$ index is defined as $$GA(G)=\sum\limits_{uv\in E(G)}{\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}}$$, where d$$_{u}$$ denotes the degree of a vertex $$u$$ in $$G$$. In this paper, we obtain the general formula for the geometric arithmetic $$GA$$ index for certain trees and unicyclic graphs with application such as Alkanes, Isomerism of Alkanes and more classes of Cycloalkanes.

### On some characterization of nearly Hall S-semiembedded subgroups of finite groups

JPRM-Vol. 17 (2021), Issue 2, pp. 79 – 84 Open Access Full-Text PDF
Iftikhar Ali, Abid Mahboob, Taswer Hussaain, Faryal Chaudhry
Abstract: Let K be a subgroup. Then K is known as partially Hall s-semiembedded subgruop in $$\Game$$ if for K$$\mathcal{T}$$ is s-semi permutable in $$\Game$$ and K $$\cap \mathcal{T} \leq$$ $$K_{\tilde{s} \Game}$$ where $$K _{\tilde{s} \Game}$$ generated by all those subgroups of K which are Hall s-semiembedded in $$\Game$$, there exists a normal subgroup $$\mathcal{T}$$ of G. In this paper, we investigate the notion of partially Hall $$S$$-semi embedded subgroups on the structure of finite group $$\Game$$. We obtain some new criteria related to the $$p$$-nilpotency and super solubility of a finite group. Some earlier results about formations are also generalized by our results.

### 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.