### Connective eccentricity index of certain path-thorn graphs

JPRM-Vol. 1 (2018), Issue 1, pp. 87 – 99 Open Access Full-Text PDF
M. Javaid, M. Ibraheem, A. A. Bhatti
Abstract: Let $$G$$ be a simple connected graph with $$V (G)$$ and $$E(G)$$ as the vertex set and edge set respectively. A topological index is a numeric quantity by which we can characterize the whole structure of a molecular graph or a network to predict the physical or chemical activities of the involved chemical compounds in the molecular graph or network. The connective eccentricity index of the graph $$G$$ is defined as $$ξ^{ce}(G) = \sum_{v∈G}\frac{d(v)}{e(v)}$$, where $$d(v)$$ and $$e(v)$$ denote the degree and eccentricity of the vertex $$v ∈ G$$ respectively. In this paper, we compute the connective eccentricity index of the various families of the path-thorn graphs and present the obtained results with the help of suitable mathematical expressions consisting on various summations. More precisely, the computed results are general extensions of the some known results.

### Solving differential equations by wavelet transform method based on the mother wavelets & differential invariants

JPRM-Vol. 1 (2018), Issue 1, pp. 74 – 86 Open Access Full-Text PDF
Hamid Reza Yazdani, Mehdi Nadjafikhah, Megerdich Toomanian
Abstract: Nowadays, wavelets have been widely used in various fields of science and technology. Meanwhile, the wavelet transforms and the generation of new Mother wavelets are noteworthy. In this paper, we generate new Mother wavelets and analyze the differential equations by using of their corresponding wavelet transforms. This method by Mother wavelets and the corresponding wavelet transforms produces analytical solutions for PDEs and ODEs.

### K Banhatti and k hyper banhatti indices of the line graphs of h-pantacenic nanotubes

JPRM-Vol. 1 (2018), Issue 1, pp. 62 – 73 Open Access Full-Text PDF
Abdur Rehman, Muhammad Nawaz, Waqas Nazeer, Wei Gao
Abstract: Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. Topological indices are used for example in the development of quantitative structureactivity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. The aim of this report is to compute the first and second K Banhatti indices of the Line Graphs of H-Pantacenic Nanotubes. We also compute the first and second K hyper Banhatti indices of the Line Graphs of H-Pantacenic Nanotubes.

### Bounds of F-index for unicyclic graphs with fixed pendent vertices

JPRM-Vol. 1 (2018), Issue 1, pp. 51 – 61 Open Access Full-Text PDF
M. Javaid, Maqsood Ahmad, M. Hussain, W.C. Teh
Abstract: Furtula and Gutman [J. Math. Chem., 53 (4) (2015), 1184- 1190] reinvestigated the $$F$$-index as a sum of cubes of the degrees of all the vertices in a chemical graph and proved its various properties. A connected graph with equal order and size is called unicyclic graph, where order is number of vertices and size is number of edges. In this paper, we characterize the extremal graphs in a family of graphs called by unicyclic graphs with fixed number of pendent vertices. We also investigate the bound on $$F$$-index in the same family of graphs i.e $$4(2n + 3α) ≤ F(G) ≤ 8n + α(α + 2)(α + 3)$$ for each $$G ∈ \mathcal{U}_{n}^{ α}$$, where $$\mathcal{U}_{n}^{ α}$$ is a class of all the unicyclic graphs such that the order of each graph is $$n$$ with $$α$$ pendent vertices.

### A new paradigm for increasing the continuity of subdivision schemes

JPRM-Vol. 1 (2018), Issue 1, pp. 37 – 50 Open Access Full-Text PDF
Abstract: Subdivision schemes having high continuity are always required for designing of smooth curves and surfaces. In this paper, we present a paradigm to generate a family of binary approximating subdivision schemes with high continuity based on probability distribution. The analysis and convexity preservation of some members of the family are also presented. Subdivision schemes give skewed behavior on convex data due to probability parameter.

### New twelfth order algorithms for solving nonlinear equations by using variational iteration technique

JPRM-Vol. 1 (2018), Issue 1, pp. 24 – 36 Open Access Full-Text PDF
Muhammad Nawaz, Amir Naseem, Waqas Nazeer
Abstract: In this paper, we proposed three new algorithms for solving non-linear equations by using variational iteration technique. We discuss the convergence criteria of our newly developed algorithms. To demonstrate the efficiency and performance of these methods, several numerical examples are given which show that our generated methods are best as compared to Newton’s method, Halley’s method, Househ¨older’s method and other well known iterative methods. The variational iteration technique can be used to suggest a wide class of new iterative methods for solving a system of non-linear equations.

### Invariance of metric dimension of subdivision of the crown graph

JPRM-Vol. 1 (2018), Issue 1, pp. 18 – 23 Open Access Full-Text PDF
Abdul Rauf Nizami
Abstract: Metric dimension has applications in several areas including computer navigation, chemistry, and image processing. In this paper we give metric dimension of the crown graph $$S_n$$, $$n ≥ 3$$, and show that it remains invariant under any of its subdivision.
### Generalized $$\xi$$-rings
Abstract: Let $$R$$ be a ring with center $$C(R)$$. A ring $$R$$ is called a ξring if, for any element $$x ∈ R$$, there exists an element $$y ∈ R$$ such that $$x − x^2y ∈ C(R)$$. In Proc. Japan Acad. Sci., Ser. A – Math. (1957), Utumi describes the structure of these rings as a natural generalization of the classical strongly regular rings, that are rings for which $$x = x^2 y$$. In order to make up a natural connection of $$ξ$$-rings with the more general class of von Neumann regular rings, that are rings for which $$x =xyx$$, we introduce here the so-called generalized $$ξ$$-rings as those rings in which $$x − xyx ∈ C(R)$$. Several characteristic properties of this newly defined class are proved, which extend the corresponding ones established by Utumi in these Proceedings (1957).