A Lower Bound for the First Hyper-Zagreb Index of Trees with given Roman Domination Number
JPRM-Vol. 21 (2025), Issue 1, pp. 49 – 54 Open Access Full-Text PDF
W. Ali, M. N. Husin
Abstract: In graph theory, the first Hyper-Zagreb index HM1(G) is calculated by summing the squares of the degrees of adjacent vertices u and v in molecular graphs. A Roman dominating function (RDF) on a graph G is a function z : V(G) → {0, 1, 2}, where V(G) is the vertex set, with the requirement that for each vertex v with z(v) = 0, there exists an adjacent vertex u such that z(u) = 2. The Roman domination number (RDN) denoted as ζR(G) and represents as the minimum total weight of all vertices under an RDF, and it plays a significant role in network analysis. In this paper, we present a new lower bound for the HZ1(T ) for trees T with order n and ζR(T ). These findings enhance our understanding of tree structures, providing chemists with a valuable tool for analyzing molecular stability and reactivity. By establishing mathematical bounds on the HZ1(T ), this research supports more precise predictions of molecular properties and aids in efficient experimental planning in chemical graph theory.
Exponential (h,m)-Convex Functions, Basic Results and Hermite-Hadamard Inequality
JPRM-Vol. 21 (2025), Issue 1, pp. 40 – 48 Open Access Full-Text PDF
Muhammad Ajmal, Muhammad Rafaqat
Abstract: This paper explores the extension of the Hermite-Hadamard inequality to exponential (h,m)-convex functions, particularly within the framework of Caputo fractional integrals. Traditional calculus often falls short in adequately modeling systems with memory and non-local interactions, which are prevalent in various scientific and engineering fields. By incorporating Caputo fractional calculus, this work addresses complex dynamic systems that exhibit memory effects, a common characteristic in materials science, financial mathematics, and thermal physics. We present a series of new theoretical results including basic properties and integral inequalities of exponential (h,m)-convex functions, alongside their fractional counterparts. Further, we provide rigorous proofs of the Hermite-Hadamard inequality in both classical and fractional settings, demonstrating its utility in estimating bounds for real-world applications. The paper concludes with a detailed discussion on the practical implications of these findings in optimizing financial models, designing advanced materials, and engineering efficient thermal systems. Our results not only extend the classical understanding of convexity and its applications but also pave the way for future research in fractional calculus and its integration into applied mathematics.
On a class of abstract fourth-order differential equations set on cusp domain
JPRM-Vol. 21 (2025), Issue 1, pp. 22 – 39 Open Access Full-Text PDF
Naceur Cheghloufa, Belkacem Chaouchi, Marko Kostic’, Fatiha Boutaous
Abstract: In this work, we concentrate on a boundary value problem set on a singular domain containing a cuspidial point. In our study, we obtain some existence and maximal regularity results. Our strategy is based on the study of a boundary value problems for a class of the complete abstract fourth-order differential equations involving fractional powers of unbounded linear operators.
Deterministic and stochastic approaches for a fat receptor-breast cancer model with crossover effects
JPRM-Vol. 21 (2025), Issue 1, pp. 1 – 21 Open Access Full-Text PDF
Maroua Amel Boubekeur
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.