This paper initiates a study on a new optimization problem with regards to graph completion. A new iterative procedure called Marcello’s completion of a graph is defined. For graph \(G\) of order \(n\) the graphs, \(G_1,G_2,\dots,G_k\) are obtained in accordance to the Marcello rule. If for smallest \(k\) the resultant graph \(G_k \cong K_n\) then the Marcello number of a graph \(G\) denoted by \(\varpi(G)\) is equal to \(\varpi(G) = k\). By convention \(\varpi(K_n) = 0\), \(n \geq 1\). Certain introductory results are presented.

The atom-bond sum-connectivity \((ABS)\) matrix of a graph \(G\) is the square matrix of order \(n\), whose \((i,j)\)-entry is equal to \(\sqrt{1-\frac{2}{d_i+d_j}}\) if the \(i\)-th vertex and the \(j\)-th vertex of \(G\) are adjacent, and \(0\) otherwise, where \(d_i\) is the degree of the \(i\)-th vertex of \(G\). The \(ABS\) spectral radius of \(G\) is the largest eigenvalue of the \(ABS\) matrix of \(G\). Recently, we studied the extremal problem for the \(ABS\) spectral radii of trees and unicyclic graphs, determining which structures achieve the maximum and minimum values. In this paper, the unicyclic graphs and bicyclic graphs with the first two largest \(ABS\) spectral radii are characterized.

The increasing prevalence of electric vehicles (EVs) in urban logistics presents challenges such as route planning, energy constraints, and demand management. EVs’ limited range, charging requirements, and sensitivity to traffic conditions necessitate advanced optimization strategies. Fleet management systems are thus evolving into intelligent, modular platforms that not only plan delivery tasks but also interact with real-time data and respond to dynamic disruptions. Among these, traffic congestion remains a critical factor that can severely affect route reliability and lead to time window violations. In this study, a modular fleet management system architecture is proposed, capable of real-time monitoring, dynamic rerouting, and traffic-aware decision-making. The system introduces a standardized data structure called the Routing Markup Language (RML), which formalizes the communication between components and supports various route outputs including simulation and vehicle-level execution. Adaptive Large Neighborhood Search (ALNS) is applied for route planning using real-world order data from a water distribution company operating in the Büyükdere district of Eskişehir. The system also features a dynamic reassignment mechanism that responds to vehicle failure scenarios, ensuring continued operation with minimal disruption. Traffic scenarios are evaluated through the Simulation of Urban Mobility (SUMO) environment to assess route robustness under varying conditions. The proposed approach integrates routing optimization, dynamic disruption handling, and simulation-supported fleet monitoring into a cohesive system, offering a responsive and data-driven solution for sustainable urban logistics.

This study investigates the effects of velocity slip and convective boundary conditions on heat transfer and entropy generation in steady magnetohydrodynamic flow of a viscous, incompressible, electrically conducting fluid with internal heat generation/absorption, offering conditions relevant to microchannel cooling, porous heat exchangers, and energy system thermal management. The governing equations were transformed into coupled ordinary differential equations and solved analytically using the method of undetermined coefficients. The analytical solutions showed strong agreement with existing results, validating the model. Parametric analyses, supported by MATLAB visualizations, examined the influence of the magnetic field, slip coefficients, Biot number, and other parameters on flow, temperature distribution, and thermodynamic irreversibility. Results indicate that velocity decreases with increasing suction, magnetic intensity, and upper-wall slip, while temperature diminishes with higher Peclet number or injection velocity. Entropy generation is primarily governed by viscous and Joule dissipation, whereas wall convection and slip act as controlling mechanisms. The Bejan number analysis reveals that heat-transfer irreversibility predominates at higher magnetic parameters, while larger slip and Biot numbers enhance viscous effects and lower Bejan values. These findings have the potential to offer practical guidelines for designing efficient porous-channel cooling system components, particularly where control over wall slip and convective heat exchange is critical to minimizing energy loss and enhancing thermal performance.

The world continues to experience rising levels of crime, particularly in regions affected by socioeconomic disparity and structural inequality. To better understand and control these dynamics, we develop a nonlinear dynamical system of ordinary differential equations describing the evolution of crime within a population. The model divides the total population into five interacting compartments: \(\mathcal{S}_1(t)\) (not-at-risk individuals), \(\mathcal{S}_2(t)\) (at-risk individuals), \(\mathcal{C}(t)\) (active criminals), \(\mathcal{H}(t)\) (habitual offenders who are resistant to rehabilitation), and \(\mathcal{R}(t)\) (rehabilitated or reformed individuals). The influence of key behavioural transition parameters—notably the crime initiation rate \((\alpha)\) and the rate of recovery from the at-risk group \((\beta)\)—on the temporal evolution of each compartment is examined using numerical simulations. Line and contour plots demonstrate that increasing \(\alpha\) enhances the recruitment of at-risk individuals into criminal activity, thereby expanding both the criminal \((\mathcal{C})\) and habitual \((\mathcal{H})\) populations. In contrast, higher \(\beta\) values promote reintegration and reduce the size of the at-risk group \((\mathcal{S}_2)\). These results emphasize the significance of prevention-based interventions (reducing \(\alpha\)) and rehabilitation-oriented strategies (enhancing \(\beta\)) in curbing persistent crime. Furthermore, the basic reproduction number \((\mathcal{R}_c)\) is derived using the next-generation matrix approach to serve as a threshold indicator for crime persistence. Analytical and graphical sensitivity analyses reveal that \(\mathcal{R}_c\) is strongly influenced by the crime transmission rate \((\beta)\), the recruitment fraction into the at-risk class \((p)\), the natural exit rate \((\mu)\), the conviction rate \((\sigma)\), and the rate of progression to habitual criminality \((\eta)\). Contour and three-dimensional surface plots identify parameter regimes for which \(\mathcal{R}_c < 1\) (crime eradication) and \(\mathcal{R}_c > 1\) (crime persistence). The study concludes that reducing recruitment into at-risk groups, increasing conviction and natural exit rates, and minimizing habitual offender influence can effectively suppress criminal propagation, providing a quantitative foundation for evidence-based crime mitigation policies.

This paper presents a thorough investigation of Laplace transforms for a wide range of probability distributions, including both classical and generalized forms. Analytical derivations are complemented with extensive numerical examples and graphical simulations to validate the theoretical results. Applications in reliability theory and system availability are discussed, highlighting the practical significance of the findings for modeling and performance evaluation of complex systems.

Pre-exposure prophylaxis (PrEP) is an effective strategy for HIV prevention, offering individual protection and broader public health benefits. Enrollment in PrEP programs not only provides access to HIV prevention but also serves as a strategic entry point for the diagnosis of other sexually transmitted infections (STIs). Because participation requires regular HIV testing and routine STI screening (e.g., for syphilis), PrEP implementation facilitates early detection and treatment of coexisting infections, strengthening integrated sexual health surveillance and control efforts. We developed a mathematical model capturing syphilis dynamics, incorporating PrEP as a mechanism for diagnosis and treatment engagement. The model considers coinfection via high-risk sexual contact, partial protection of PrEP (HIV but not syphilis), and diagnostic pathways linked to PrEP program entry. Independent analysis of syphilis (without PrEP) established population persistence, basic reproduction numbers, and stability of disease-free and endemic equilibria. Integrating PrEP, we derived conditions under which PrEP-related parameters—particularly diagnostic access—positively influence syphilis transmission dynamics. Sensitivity analysis showed that higher PrEP adherence reduces reproduction numbers for syphilis and coinfection. Computational simulations using literature-based parameters confirmed these findings: increased PrEP use and lower discontinuation rates decreased new infections and improved treatment outcomes. These results highlight the role of PrEP in improving the detection and treatment of syphilis and HIV–syphilis coinfection.