The elliptic Sombor index ESO( G ) = ∑ uv ∈ E ( d ( u ) + d ( v ))√ d ( u ) 2 + d ( v ) 2 is a recently introduced degree-based topological index. This paper corrects several inaccuracies in the original paper [ RAIRO-Oper. Res. 59 (2025) 2201–2212] and supplies valid proofs in place of the flawed ones. Moreover, we present new spectral bounds and sharp comparisons with several classical degree-based topological indices.
The topology of interconnection networks is fundamental to the performance and reliability of parallel and distributed systems. Typically, an interconnection network is modeled as a graph, and connectivity of a graph serves as a critical parameter for measuring network reliability. The $h$-extra $r$-component vertex (resp. edge) connectivity of a graph~$G$, denoted by $c\kappa_{r}^{h}(G)$ (resp. $c\lambda_{r}^{h}(G)$), is the minimum number of vertices (resp. edges) whose removal results in a disconnected graph with at least $r$ components, in which each component contains at least $h+1$ vertices. In this paper, we investigate the hierarchical pancake network~$\mathcal{HP}_n$, a highly symmetric and recursively structured Cayley-type topology, and show that $c\kappa^1_3(\mathcal{HP}_n) = 4n - 6$ and $c\lambda^1_3(\mathcal{HP}_n) = 4n - 6$ for $n \ge 4$.
The Walrus Optimization Algorithm (WaOA) is a recently developed meta-heuristic inspired by the natural and social behaviour of walruses. This study presents two multi-objective extensions of WaOA: the Multi-ObjectiveWalrus Optimization Algorithm (MO-WaOA) and the Multi-Objective Chaotic Walrus Optimization Algorithm (MO-CWaOA). These methods serve as effective adaptations of WaOA for multi-objective problems. MO-CWaOA enhances solution diversity by integrating chaotic-based mechanisms into the optimization process. Different chaotic maps are explored by replacing random elements with chaotic sequences, aiding in the balance between exploration and exploitation. A statistical comparison between MO-CWaOA and its non-chaotic counterpart (MO-WaOA) is conducted to assess the impact of the chaotic operator. The proposed techniques are evaluated using seven well-established benchmark functions and five standard constrained engineering design benchmarks. Their performance is compared with six established multi-objective optimization algorithms: Multi-Objective Evolutionary Algorithm Based on Decomposition (MOEA/D), Multi-Objective Multi-Verse Optimization (MOMVO), Multi-Objective Artificial Vultures Optimization Algorithm (MOAVOA), Non-dominated Sorting Genetic Algorithm III (NSGAIII), MOEA/D with Detection-and-Escape strategy (MOEA/D-DAE), and Strength Pareto Evolutionary Algorithm 2 with Constraint-Domination Principle (SPEA2-CDP), using four performance metrics. The obtained results, both qualitative and quantitative, demonstrate competitive performance for the proposed approaches.
The widespread adoption of agricultural carbon labeling is critical for sustainability but faces impediments from environmental uncertainties and inherent cost-benefit misalignments. This study constructs a tripartite stochastic evolutionary game model to analyze the behavioral dynamics of producers, consumers, and local governments. By incorporating Gaussian white noise into replicator dynamic equations and utilizing the stochastic Taylor expansion method for numerical approximations, the research evaluates the stability and resilience of the governance system, grounded in the specific context of the carbon-labeled agricultural initiative in Xiangshui, China. Numerical simulations identify the critical conditions for institutional stability and yield four primary findings. First, analyzing the initial strategy selection reveals a persistent supply and demand mismatch because green production costs consistently exceed market premiums. Second, production subsidies delay producer exit while weakening supervisory capacity, whereas consumer incentives accelerate green transitions but fail to stabilize the system due to insufficient producer profitability. Third, raising enforcement standards increases the administrative cost per inspection and forces authorities to reduce supervision frequency. Finally, setting policy targets beyond local administrative limits causes severe regulatory failure, proving that institutional stability depends on matching policy goals with practical execution capacity rather than relying solely on market incentives.The widespread adoption of agricultural carbon labeling is critical for sustainability but faces impediments from environmental uncertainties and inherent cost-benefit misalignments. This study constructs a tripartite stochastic evolutionary game model to analyze the behavioral dynamics of producers, consumers, and local governments. By incorporating Gaussian white noise into replicator dynamic equations and utilizing the stochastic Taylor expansion method for numerical approximations, the research evaluates the stability and resilience of the governance system, grounded in the specific context of the carbon-labeled agricultural initiative in Xiangshui, China. Numerical simulations identify the critical conditions for institutional stability and yield four primary findings. First, analyzing the initial strategy selection reveals a persistent supply and demand mismatch because green production costs consistently exceed market premiums. Second, production subsidies delay producer exit while weakening supervisory capacity, whereas consumer incentives accelerate green transitions but fail to stabilize the system due to insufficient producer profitability. Third, raising enforcement standards increases the administrative cost per inspection and forces authorities to reduce supervision frequency. Finally, setting policy targets beyond local administrative limits causes severe regulatory failure, proving that institutional stability depends on matching policy goals with practical execution capacity rather than relying solely on market incentives.
Growing awareness of climate change and environmental issues, purchasing behaviour of the customers. Nowadays, designing of inventory policy affected due to the presence of sustainable products, electric vehicles, etc., in the market. Further, advance payment scheme also affected the inventory policy designed by decision-maker as this is one of the financing schemes offered by non-financial institution. Focussing, all these challenging issues, this paper presents a green inventory model that includes advanced payment policy and preservation technology investment. In the model, decision-maker designed the inventory policy considering carbon emission dependent demand. Further, preservation technology investment reduces the rate of deterioration. A credit period has been offered by supplier to retailer under advance payment scheme to pay purchased cost. The objective of the present paper is to reduce the impact of carbon emissions and minimize the business investment of retailers. In the model, two different scenarios: one where advance payment scheme is considered and other where is no scheme of advance payment is offered by supplier have been discussed. Under each case, two subcases are also derived based on preservation technology investment either applied or relaxed. Numerical examples have been illustrated to test the practical applicability of the model. Model's sensitivity depending upon key parameters and managerial insight have been also presented. Numerical study reveals that cost is at minimum when no advance payment scheme is offered and there is no investment made in preservation technology. More investment in preservation technology results 25% reduction in deterioration rate.
