Journal of Prime Research in Mathematics

Abstract: In this paper, the dynamics of a fat receptor-breast cancer model have been investigated by employing the deterministic and stochastic approaches. The existence of the endemic equilibrium, positivity of solutions and the calculation of the reproduction number are examined for the deterministic model and also the existence uniqueness of the stochastic model is discussed. Then, we will examine the crossover tendencies of the deterministic-stochastic model with the help of piecewise differential operators that take into account stochastic and power law processes followed by generalized Mittag-Leffler functions have been investigated. We employ a numerical scheme based on Newton polynomial to solve the deterministic-stochastic tumor growth model with fractional differential operators numerically. The graphical representations are simulated for different values of fractional order and the crossover tendencies of the deterministic-stochastic model are observed during the simulations.

Abstract: In this paper, we propose a new type of module by focusing on the second cosingular submodule of a module. We define a module M as weak T-dual Rickart if, for any homomorphism φ ∈ EndR(M), the submodule φ(Z̄²(M)) lies above a direct summand of M. We prove that this property is inherited by direct summands of M. We also introduce weak T-dual Baer modules and provide a complete characterization of such modules where the second cosingular submodule is a direct summand. Furthermore, we present a characterization of (semi)perfect rings in which every (finitely generated) module is weak T-dual Rickart.

Abstract: We provide a thorough examination of the graph energies in regular graphs that arise from the vertex duplication process in this paper. Understanding the numerous structural components of graphs requires understanding the thought of graph energy, known as a measurement obtained by computing eigenvalues the adjacency matrix of a graph. We derived generalized closed-form expressions for a number of important energy metrics, such as minimum degree, energy maximum degree energy, first Zagreb energy, second Zagreb energy and degree square sum energy, utilizing proficient algebraic graph theory techniques and eigenvalue spectrum analysis. Our work emphasizes on vertex duplication techniques and the impact they have on these energy metrics, primarily on regular graphs, a basic class of graphs where every vertex has the same degree. The resulting formulations offer further explanations for the behavior and attributes of these energy functions within the framework of regular graphs, providing a more comprehensive knowledge of how these operations affect the structural complexity of the graph. These findings greatly expand the conceptual model of graph energy and have potential uses in fields like combinatorics, chemistry, and network analysis where the energy models of graphs are extensively employed.

Abstract: The aim of this paper is to investigate reflexive edge strength in graph theory, defined as the specialized area of an edge that is irregularly labeled, where both vertices and edges are labeled. The reflexive edge strength, res(G), is the minimal value of k for which the sum of weights of any two different edges in agraph is distinct. In this paper, reflexive edge strength of b-subdivided ladder graphs and the triangular ladder graph studied.

Abstract: A directed Toeplitz graph T_n⟨a₁, . . . , a_p; b₁, . . . , b_q⟩ with vertices 1, 2, . . . , n, where the edge (i, j) occurs if and only if j − i = a_s or i − j = b_t for some 1 ≤ s ≤ p and 1 ≤ t ≤ q, is a digraph whose adjacency matrix is a Toeplitz matrix. In this paper, we study hamiltonicity in directed Toeplitz graphs having increasing edges of length 1, 3 and 7, only.

Abstract: This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatics and definitions of addition and multiplication of points on a line in this plane, and for skew field properties. In this paper are studied, properties and results related to the some transforms for cross ratio for 4-points, in a line, which we divide into two categories, Invariant and Preserving transforms for cross ratio. The results in this paper are (1) the cross-ratio of four points is Invariant under transforms: Inversion, Natural Translation, Natural Dilation, Mobi¨us Transform, in a line of Desargues affine plane. (2) the cross-ratio of four points is Preserved under transforms: parallel projection, translations and dilation’s in the Desargues affine plane.