Graph labeling is an important and rapidly developing area in graph theory due to its numerous applications in communication networks, frequency assignment, channel allocation, and coding theory. Among the various labeling methods, radial radio labeling has gained attention because of its connection with distance-based constraints and graph structure. In this paper, we study the radial radio labeling of certain connected subgraphs derived from complete graphs. We introduce the concepts of radial radio number and radial radio sequence and examine their behavior for specific graph classes obtained from complete graphs through edge deletions. The main objective of this work is to determine the radial radio sequence of connected graphs formed by deleting a perfect matching and an edge covering from the complete graph. Using fundamental graph-theoretic techniques and structural analysis, we derive exact values for the radial radio sequence of these graph families. The obtained results provide insight into the influence of graph structure and neighborhood properties on radial radio labeling. This study extends existing research on distance-based graph labeling and contributes to a better understanding of how structural modifications in graphs affect labeling parameters, which may further support applications in communication network optimization and interference reduction problems.
Breast cancer remains one of the leading causes of cancer-related mortality worldwide, and its complex interaction with the immune system and therapeutic interventions presents significant challenges for mathematical modeling. Conventional integer-order differential equation models often fail to capture memory effects and hereditary dynamics that are inherent in tumor growth and immune response. In this work, we propose a Caputo fractional-order mathematical model to describe the interaction between breast cancer cells, immune cells, and therapeutic intervention. The model incorporates the fractional-order parameter to account for memory effects in biological tissues and the long-term influence of past states on disease progression. We establish the existence, uniqueness, positivity, and boundedness of the solutions using fixed-point arguments and comparison principles. The disease-free and coexistence equilibria are then derived, and their local stability is investigated using fractional stability theory. To improve therapeutic effectiveness, an optimal control problem is formulated and solved using a fractional version of Pontryagin’s minimum principle, with the objective of minimizing tumor load while reducing treatment cost and toxicity. Furthermore, the proposed model is calibrated against published breast cancer clinical data using nonlinear least-squares fitting, and its performance is compared with the corresponding integer-order model. Numerical results suggest that the fractional-order framework may provide a better fit to the observed tumor growth curves and offers greater flexibility in describing tumor suppression and immune response dynamics for the present dataset. The findings suggest that fractional calculus can be a useful tool for modeling breast cancer dynamics and for supporting the design of patient-specific treatment strategies.
This study investigates the unsteady Magnetohydrodynamic (MHD) flow and heat transfer characteristics of a nonlinear third-grade non-Newtonian fluid with temperature-dependent viscosity and nonlinear thermal radiation, which are commonly encountered in advanced engineering systems such as polymer processing, metallurgical operations, liquid-metal cooling technologies, and energy conversion devices. Owing to its viscoelastic nature, which deviates from classical Newtonian assumptions, the fluid behavior is described by coupled nonlinear momentum and energy equations incorporating magnetic field effects, viscous dissipation, nonlinear shear contributions, variable thermal conductivity, and wall suction. The governing equations are non-dimensionalised to identify the key controlling physical parameters and solved numerically using an explicit finite difference scheme, enabling a detailed parametric analysis of the effects of magnetic strength, nonlinear material parameters, radiation intensity, and viscosity variation on velocity and temperature distributions within the boundary layer. The results indicate that increasing magnetic field strength suppresses fluid motion through Lorentz force effects, thereby thinning the momentum boundary layer and providing an effective mechanism for electromagnetic flow control. Additionally, nonlinear rheological parameters significantly alter momentum transport, while radiative heat transfer and viscous dissipation elevate the thermal energy within the fluid, and variations in thermal conductivity strongly influence heat diffusion and temperature gradients. These findings offer valuable design insights for enhancing flow regulation and thermal performance in industrial systems involving electrically conducting non-Newtonian fluids operating under magnetic fields and high-temperature conditions.
The purpose of this paper is to consider a stochastic differential equation guiding X a continuous d-dimensional diffusion process, the coefficients of which depending on the pair (X, M) , where M is the running supremum of the first component X 1 . As an application of our work, we could think of a firm the activity of which is characterized by a set of processes ( X 1 ,···, X d ). But one of them, for instance X 1 , could be linked to an alarm. Such a ( d + 1)- dimensional process (X, M) could present a crucial interest in this case where M could be an alarm. Indeed, the possibility of an alarm at time t namely the event {∃s ≤ t : X 1 s > u} is identical to the event {M t > u } > when the specific (and dangerous) threshold u is exceeded. This means that the law of M is closely linked to the law of the hitting time when X 1 reaches such a dangerous level u . Here is proved that, for all positive real number t; the law of the ( d +1)-dimensional random vector ( X t , M t ) admits a density with respect to the Lebesgue measure. The solution of such a stochastic differential equation is built using a recursion method. The existence of the density of the law of ( X , M ) is based on Malliavin calculus. This density is solution of a partial differential equation in a weak sense. Moreover, such a recursive construction will allow to build simulated solutions. So finally such a tool could allow us to build an alarm system to detect the hitting time when the alarm could occur.
An edge coloring of a simple, connected, undirected graph G is an assignment of color labels to the edges of G such that no two adjacent edges are assigned the same color. The first-fit or greedy edge coloring algorithm colors edges sequentially by assigning the smallest available positive integer color to each edge. The Grundy edge chromatic number, denoted by Γ'( G ), captures the worst-case performance of this greedy approach, defined formally as the maximum number of colors utilized by the greedy edge coloring algorithm across all possible orderings of the edge set E ( G ). Determining Γ'( G ) is known to be computationally hard in general, prompting significant research into deriving exact values and structural bounds for specialized graph families and graph operations. In this paper, we explore the Grundy edge chromatic number for several key topological families and structured composite graphs. First, we focus on the shadow graph operation, denoted D 2 ( G ), which duplicates the vertex set of G and interconnects vertices based on their neighborhood adjacencies. We establish exact explicit formulas for the Grundy edge chromatic number of shadow graphs constructed from foundational graph structures, including paths P n , cycles C n , sunlet graphs S n , and tadpole graphs T n , k . Second, we extend our investigation to complex wheel-like and symmetric network architectures. Specifically, we evaluate the structural properties and derive exact closed-form expressions for the Grundy edge chromatic number of closed helm graphs CH n , double wheel graphs DW n , and djembe graphs D n . For each considered graph class, we construct explicit edge orderings that force the greedy algorithm to attain its theoretical upper bound, thereby matching the structural upper bounds derived from degree conditions and neighborhood characteristics. These exact results provide comprehensive insights into the performance bounds of greedy edge-labeling heuristics in interconnection networks, expanding the current understanding of greedy coloring parameters in graph theory.
