Journal of Prime Research in Mathematics

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.

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.

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.

Abstract: The core idea of this paper is to provide the Opial-type inequalities for Hadamard fractional integral operator and fractional integral of a function with respect to an increasing function \(g\). Moreover, related extreme cases and counter part of our main results are also given in the paper.

Abstract: Topological indices are numerical values that correlate the chemical structures with physical properties. In this article, we compute some reverse topological indices namely reverse Atom-bond connectivity index and reverse Geometric-arithmetic index for four different types of Benzenoid systems.

Abstract: It is known that in mathematical literature one of important questions of spectral theory of operators is to describe spectrum of diagonal block matrices in the direct sum of Banach spaces with the spectrums of their coordinate operators. This problem has been investigated in works [1] and [2]. Also for the singular numbers similar investigation has been made in [3]. In this paper the analogous question is researched. Namely, the relationships between \(\epsilon\)-determinat spectrums of the diagonal block matrices and their block matrices are investigated. Later on, some applications are given.