# 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 coindex and Wiener polarity index of Jahangir graphs

JPRM-Vol. 17 (2021), Issue 2, pp. 8 – 14 Open Access Full-Text PDF
Abstract: The Leap Zagreb coindices and the Wiener polarity index are examples of less frequently studied topological indices that are based on the number of vertices lying at second and third distances from a given vertex. These indices have applications in QSAR and QSPR studies. In this paper, we obtain leap Zagreb coindices and the Wiener polarity index of a rotationally symmetric subfamily of circulant graphs known as Jahangir graphs.

### Parity labeling in Signed Graphs

JPRM-Vol. 17 (2021), Issue 2, pp. 1 – 14 Open Access Full-Text PDF
Mukti Acharya, Joseph Varghese Kureethara
Abstract: Let $$S=(G, \sigma)$$ be a signed graph where $$G=(V, E)$$ is a graph called the underlying graph of $$S$$ and $$\sigma:E(G) \rightarrow \{+,~-\}$$. Let $$f:V(G) \rightarrow \{1,2,\dots,|V(G)|\}$$ such that $$\sigma(uv)=+$$ if $$f(u)$$ and $$f(v)$$ are of same parity and $$\sigma(uv)=-$$ if $$f(u)$$ and $$f(v)$$ are of opposite parity. The bijection $$f$$ induces a signed graph $$G_f$$ denoted as $$S$$, which is a parity signed graph. In this paper, we initiate the study of parity labeling in signed graphs. We define and find `rna’ number denoted as $$\sigma^-(S)$$ for some classes of signed graphs. We also characterize some signed graphs which are parity signed graphs. Some directions for further research are also suggested.

### A cubic trigonometric B-spline collocation method based on Hermite formula for the numerical solution of the heat equation with classical and non-classical boundary conditions

JPRM-Vol. 17 (2021), Issue 1, pp. 84 – 94 Open Access Full-Text PDF
Abstract: In this article, a new trigonometric cubic B-spline collocation method based on the Hermite formula is presented for the numerical solution of the heat equation with classical and non-classical boundary conditions. This scheme depends on the standard finite difference scheme to discretize the time derivative while cubic trigonometric B-splines are utilized to discretize the derivatives in space. The scheme is further refined utilizing the Hermite formula. The stability analysis of the scheme is established by standard Von-Neumann method. The numerical solution is obtained as a piecewise smooth function empowering us to find approximations at any location in the domain. The relevance of the method is checked by some test problems. The suitability and exactness of the proposed method are shown by computing the error norms. Numerical results are compared with some current numerical procedures to show the effectiveness of the proposed scheme.

### Partition dimension of generalized Peterson and Harary graphs

JPRM-Vol. 17 (2021), Issue 1, pp. 84 – 94 Open Access Full-Text PDF
Abstract: The distance of a connected, simple graph $$\mathbb{P}$$ is denoted by $$d({\alpha}_1,{\alpha}_2),$$ which is the length of a shortest path between the vertices $${\alpha}_1,{\alpha}_2\in V(\mathbb{P}),$$ where $$V(\mathbb{P})$$ is the vertex set of $$\mathbb{P}.$$ The $$l$$-ordered partition of $$V(\mathbb{P})$$ is $$K=\{K_1,K_2,\dots,K_l\}.$$ A vertex $${\alpha}\in V(\mathbb{P}),$$ and $$r({\alpha}|K)=\{d({\alpha},K_1),d({\alpha},K_2),\dots,d({\alpha},K_l)\}$$ be a $$l$$-tuple distances, where $$r({\alpha}|K)$$ is the representation of a vertex $${\alpha}$$ with respect to set $$K.$$ If $$r({\alpha}|K)$$ of $${\alpha}$$ is unique, for every pair of vertices, then $$K$$ is the resolving partition set of $$V(\mathbb{P}).$$ The minimum number $$l$$ in the resolving partition set $$K$$ is known as partition dimension ($$pd(\mathbb{P})$$). In this paper, we studied the generalized families of Peterson graph, $$P_{{\lambda},{\chi}}$$ and proved that these families have bounded partition dimension.

### Degree-based topological indices and polynomials of cellulose

JPRM-Vol. 17 (2021), Issue 1, pp. 70 – 83 Open Access Full-Text PDF
Abdul Jalil M. Khalaf, M.C. Shanmukha, A. Usha, K.C. Shilpa, Murat Cancan
Abstract: This work attempts to compute cellulose’s chemical structure using topological indices based on the degree and its neighbourhood. The study of graphs using chemistry attracts a lot of researchers globally because of its enormous applications. One such application is discussed in this work, where the structure of cellulose is considered for which the computation of topological indices and analysis of the same are carried out. A polymer is a repeated chain of the same molecule stuck together. Glucose is a natural polymer also called, Polysaccharide. The diet of the humans include fibre which contains cellulose but direct consumption of the same may not be digestible by them.

### On a third-order fuzzy difference equation

JPRM-Vol. 17 (2021), Issue 1, pp. 59 – 69 Open Access Full-Text PDF
Ibrahim Yalcinkaya, Nur Atak, Durhasan Turgut Tollu
Abstract: In this paper, we investigate the qualitative behavior of the fuzzy
difference equation
\begin{equation*}
z_{n+1}=\frac{z_{n-2}}{C+z_{n-2}z_{n-1}z_{n}}\
\end{equation*}
where $$n\in \mathbb{N}_{0}=\mathbb{N}\cup \left\{ 0\right\}$$, $$(z_{n})$$ is a sequence of positive fuzzy numbers, $$C$$ and initial conditions $$z_{-2},z_{-1},z_{0}$$ are positive fuzzy numbers.