Journal of Prime Research in Mathematics

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.

Abstract: In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. We provide explicit formulas and determinantal expressions. Our results are then applied to secondorder recurrent polynomials, presenting several relationships and identities involving Fibonacci polynomials of order 2, Lucas polynomials of order 2, classical Fibonacci polynomials, classical Lucas polynomials, Fibonacci numbers, Lucas numbers, and both kinds of Dickson polynomials. Our findings offer a unified generalization of various existing works, with several well-known results emerging as special cases.

Abstract: The manuscript explores a fractional order heat-mass transfer model for Oldroyd-B fluid on a vertical plate employing the Prabhakar derivative operator. It investigates the MHD flow of Oldroyd-B fluid induced by natural convection and the general motion of the plate. Well known integral transform i.e. Laplace transform and Tzou’s numerical inversion algorithm are employed to simulate the model under generalized constraints. Several scenarios involving the generalized motion of the plate and general boundary conditions are investigated. A comprehensive graphical analysis is conducted to investigate the control of system parameters, and valuable findings are concluded that can help to optimize and foster various processes.