Pub Date : 2026-01-08DOI: 10.1016/j.amc.2025.129937
Zhuoer An , Xinghua Liu , Gaoxi Xiao , Yu Kang , Peng Wang
An event-triggered tube model predictive control (ETT-MPC) strategy is proposed for multi-region interconnected power systems, which takes into account the uncertain interference and communication redundancy issues during frequency regulation by introducing hybrid energy storage systems (HESS). The paper investigates load frequency control of multi-region interconnected power systems with HESS units, suppressing the uncertainty of external disturbances in considered system by proposing an event-triggered tube MPC algorithm for the frequency regulation. Moreover, a robust stability criterion based on invariant sets is derived to prove the stability of the considered power system. At last, numerical examples are presented to verify that the event-triggered tube MPC method can provide the effective frequency regulation performance for the multi-region interconnected power systems.
{"title":"Load frequency control of multi-region interconnected power systems with HESS: An event-triggered tube MPC strategy","authors":"Zhuoer An , Xinghua Liu , Gaoxi Xiao , Yu Kang , Peng Wang","doi":"10.1016/j.amc.2025.129937","DOIUrl":"10.1016/j.amc.2025.129937","url":null,"abstract":"<div><div>An event-triggered tube model predictive control (ETT-MPC) strategy is proposed for multi-region interconnected power systems, which takes into account the uncertain interference and communication redundancy issues during frequency regulation by introducing hybrid energy storage systems (HESS). The paper investigates load frequency control of multi-region interconnected power systems with HESS units, suppressing the uncertainty of external disturbances in considered system by proposing an event-triggered tube MPC algorithm for the frequency regulation. Moreover, a robust stability criterion based on invariant sets is derived to prove the stability of the considered power system. At last, numerical examples are presented to verify that the event-triggered tube MPC method can provide the effective frequency regulation performance for the multi-region interconnected power systems.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129937"},"PeriodicalIF":3.4,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-08DOI: 10.1016/j.amc.2025.129938
Xuting Liang, Qiong Wang, Wei Chen
As global carbon neutrality goals drive the rapid growth of the new energy vehicle industry, accurately forecasting its sales has become a critical challenge. This paper proposes a novel fractional reverse accumulated non-equidistant grey time power model. Using fractional and reverse accumulation operators, the proposed model improves forecasting capability and the ability to capture data trends. Furthermore, its non-isometric operator effectively handles non-equidistant time series data. The model’s effectiveness is validated through simulation experiments and multiple practical case studies. Finally, the model is applied to forecast the annual sales of battery electric vehicles in China. The results predict that China’s annual sales of battery electric vehicles are expected to reach between 8.60 million and 9.10 million units by 2026, providing a comprehensive quantitative analysis for assessing market trends.
{"title":"Forecasting new energy vehicle sales using a fractional reverse accumulation non-equidistant grey time power model","authors":"Xuting Liang, Qiong Wang, Wei Chen","doi":"10.1016/j.amc.2025.129938","DOIUrl":"10.1016/j.amc.2025.129938","url":null,"abstract":"<div><div>As global carbon neutrality goals drive the rapid growth of the new energy vehicle industry, accurately forecasting its sales has become a critical challenge. This paper proposes a novel fractional reverse accumulated non-equidistant grey time power model. Using fractional and reverse accumulation operators, the proposed model improves forecasting capability and the ability to capture data trends. Furthermore, its non-isometric operator effectively handles non-equidistant time series data. The model’s effectiveness is validated through simulation experiments and multiple practical case studies. Finally, the model is applied to forecast the annual sales of battery electric vehicles in China. The results predict that China’s annual sales of battery electric vehicles are expected to reach between 8.60 million and 9.10 million units by 2026, providing a comprehensive quantitative analysis for assessing market trends.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129938"},"PeriodicalIF":3.4,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-08DOI: 10.1016/j.amc.2025.129935
Fatemeh Moaven, Mostafa Abbaszadeh
Automated arrhythmia screening from electrocardiograms (ECGs) has advanced rapidly, yet many accurate systems lack physiological interpretability. We introduce a hybrid, physics augmented pipeline that fuses 50 statistical components from principal component analysis (PCA) with two interpretable, physics informed neural network (PINN) coefficients (c1, c3) estimated per image by minimizing a simplified CLG style partial differential equation residual. The fused 52 dimensional vector is classified by a lightweight multilayer perceptron (128 → 64 → 32, ReLU, softmax). On a patient disjoint held out test set of 1,161 ECG images (class counts: F=161; M/N/Q/S/V=200 each), the hybrid model attains 95.43% accuracy and 95.00% macro F1, outperforming a PCA only baseline (92.59% accuracy; 93.00% macro F1) and a PINN only variant (18.52% accuracy). Per class gains are most pronounced for clinically challenging supraventricular (S) and ventricular (V) ectopic beats (F1: 88% → 92% and 90% → 93%, respectively). One vs rest ROC analysis shows near perfect discrimination (AUC ≈ 1.00 for F/M/N/Q and ≈ 0.99 for S/V), while confusion matrix inspection confirms reduced misclassification within the S↔V pair (net 19 → 18). Qualitative exemplars indicate that c1 (propagation/relaxation) and c3 (damping/cross coupling) inject mechanistic context that complements shape dominant PCA features, aiding separation when morphology alone is ambiguous (e.g., widened QRS supraventricular beats or narrow complex ventricular ectopy). The resulting model is compact, interpretable, and competitive with deeper vision based pipelines, suggesting a practical path toward clinically transparent ECG classification and a template for physics aware learning in biomedical signal analysis.
{"title":"Hybrid PCA-PINN framework for accurate and transparent ECG arrhythmia detection","authors":"Fatemeh Moaven, Mostafa Abbaszadeh","doi":"10.1016/j.amc.2025.129935","DOIUrl":"10.1016/j.amc.2025.129935","url":null,"abstract":"<div><div>Automated arrhythmia screening from electrocardiograms (ECGs) has advanced rapidly, yet many accurate systems lack physiological interpretability. We introduce a hybrid, physics augmented pipeline that fuses 50 statistical components from principal component analysis (PCA) with two interpretable, physics informed neural network (PINN) coefficients (<em>c</em><sub>1</sub>, <em>c</em><sub>3</sub>) estimated per image by minimizing a simplified CLG style partial differential equation residual. The fused 52 dimensional vector is classified by a lightweight multilayer perceptron (128 → 64 → 32, ReLU, softmax). On a patient disjoint held out test set of 1,161 ECG images (class counts: F=161; M/N/Q/S/V=200 each), the hybrid model attains 95.43% accuracy and 95.00% macro F1, outperforming a PCA only baseline (92.59% accuracy; 93.00% macro F1) and a PINN only variant (18.52% accuracy). Per class gains are most pronounced for clinically challenging supraventricular (S) and ventricular (V) ectopic beats (F1: 88% → 92% and 90% → 93%, respectively). One vs rest ROC analysis shows near perfect discrimination (AUC ≈ 1.00 for F/M/N/Q and ≈ 0.99 for S/V), while confusion matrix inspection confirms reduced misclassification within the S↔V pair (net 19 → 18). Qualitative exemplars indicate that <em>c</em><sub>1</sub> (propagation/relaxation) and <em>c</em><sub>3</sub> (damping/cross coupling) inject mechanistic context that complements shape dominant PCA features, aiding separation when morphology alone is ambiguous (e.g., widened QRS supraventricular beats or narrow complex ventricular ectopy). The resulting model is compact, interpretable, and competitive with deeper vision based pipelines, suggesting a practical path toward clinically transparent ECG classification and a template for physics aware learning in biomedical signal analysis.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129935"},"PeriodicalIF":3.4,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-07DOI: 10.1016/j.amc.2025.129933
Yuhui Song , Huanqing Wang , Xiaoping Liu
This study investigates the command filter-based adaptive neural predefined-time output-feedback control issue for nonlinear switched systems with arbitrary switching rule. Radial basis function neural networks (RBFNNs) are used to estimate uncertain nonlinearities, and a linear state observer is designed to estimate the unmeasurable states. Moreover, an event-triggered mechanism is utilized to alleviate the communication load. Specifically, a command filter technique is applied to tackle the computational complexity arising from the iterative differentiations of the indirect control functions. A command filter-based adaptive neural predefined-time output-feedback control strategy is formulated under the backstepping control framework, integrating the command filter control and the event-triggered mechanism. The developed control strategy guarantees that all the system signals are bounded and the tracking error converges to a little interval near origin within the predefined time. Finally, the simulation experiments reveal the validity of the devised control methodology.
{"title":"Predefined-time adaptive neural output-feedback control with filtered compensation for switched systems via event-triggered communication","authors":"Yuhui Song , Huanqing Wang , Xiaoping Liu","doi":"10.1016/j.amc.2025.129933","DOIUrl":"10.1016/j.amc.2025.129933","url":null,"abstract":"<div><div>This study investigates the command filter-based adaptive neural predefined-time output-feedback control issue for nonlinear switched systems with arbitrary switching rule. Radial basis function neural networks (RBFNNs) are used to estimate uncertain nonlinearities, and a linear state observer is designed to estimate the unmeasurable states. Moreover, an event-triggered mechanism is utilized to alleviate the communication load. Specifically, a command filter technique is applied to tackle the computational complexity arising from the iterative differentiations of the indirect control functions. A command filter-based adaptive neural predefined-time output-feedback control strategy is formulated under the backstepping control framework, integrating the command filter control and the event-triggered mechanism. The developed control strategy guarantees that all the system signals are bounded and the tracking error converges to a little interval near origin within the predefined time. Finally, the simulation experiments reveal the validity of the devised control methodology.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129933"},"PeriodicalIF":3.4,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-06DOI: 10.1016/j.amc.2025.129917
Hongji Li, Zhijun Tan
In this paper, we propose a novel singularity- and discontinuity-capturing physics-informed neural network (SDC-PINN) designed to tackle challenging time-fractional diffusion problems characterized by initial singularities and interfaces on complex curved surfaces. The SDC-PINN method incorporates a singularity-capturing feature in the temporal direction at the initial time, alongside a discontinuity-capturing feature in the spatial direction within the neural network input. This approach preserves the inherent properties of the solution, effectively and accurately capturing the solution’s sharpness as well as that of its derivatives at the interface, while adeptly resolving the initial singularity. To further address the initial singularity of the solution, we employ the classical nonuniform L2-1σ scheme in time to approximate the Caputo fractional derivative of the neural network output. Additionally, the proposed network accommodates the use of scattered training points, thereby facilitating the efficient handling of problems on more complex curved surfaces. Numerous numerical experiments are performed to assess the effectiveness and precision of the proposed SDC-PINN.
{"title":"A novel singularity- and discontinuity-capturing PINN for time-fractional diffusion equations involving initial singularities and interfaces on complex curved surfaces","authors":"Hongji Li, Zhijun Tan","doi":"10.1016/j.amc.2025.129917","DOIUrl":"10.1016/j.amc.2025.129917","url":null,"abstract":"<div><div>In this paper, we propose a novel singularity- and discontinuity-capturing physics-informed neural network (SDC-PINN) designed to tackle challenging time-fractional diffusion problems characterized by initial singularities and interfaces on complex curved surfaces. The SDC-PINN method incorporates a singularity-capturing feature in the temporal direction at the initial time, alongside a discontinuity-capturing feature in the spatial direction within the neural network input. This approach preserves the inherent properties of the solution, effectively and accurately capturing the solution’s sharpness as well as that of its derivatives at the interface, while adeptly resolving the initial singularity. To further address the initial singularity of the solution, we employ the classical nonuniform L2-1<sub><em>σ</em></sub> scheme in time to approximate the Caputo fractional derivative of the neural network output. Additionally, the proposed network accommodates the use of scattered training points, thereby facilitating the efficient handling of problems on more complex curved surfaces. Numerous numerical experiments are performed to assess the effectiveness and precision of the proposed SDC-PINN.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129917"},"PeriodicalIF":3.4,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145902637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-06DOI: 10.1016/j.amc.2025.129934
Jin You , Yan Li , Xiangyang Cao , Zhen Zhuang , Pu Ren
Optimal consensus of multi-agent systems under stochastic disturbances yields valuable insights into improving collaborative efficiency, resource utilization, and system robustness amidst uncertain environments. This study focuses on achieving optimal consensus in fractional-order multi-agent systems (FOMASs) with additive noises, an emerging research area with significant theoretical and practical implications, which deepens our understanding of system behavior and supports precise modeling and control strategy development. First, we introduce an innovative approach using a frequency distribution model to bridge fractional-order and integer-order systems, deriving sufficient conditions for Lyapunov stability. Second, we establish and address an optimal control problem tailored for stochastic Caputo fractional-order systems, laying the foundation for exploring the balance between control performance and robustness in the presence of stochastic perturbations and fractional-order dynamics. Additionally, we tackle optimal consensus control problems for stochastic FOMASs, providing sufficient conditions for achieving optimal mean square consensus through optimal control strategies and consensus conditions. The complexity of solution expressions is heightened by the presence of random variables. Finally, practical applicability is demonstrated by applying these findings to the distributed circuits and battery circuit equalization problem, thereby expanding the application potential of fractional-order systems in renewable energy.
{"title":"Distributed optimal consensus for stochastic fractional-order multi-agent systems: Frequency distribution model approach","authors":"Jin You , Yan Li , Xiangyang Cao , Zhen Zhuang , Pu Ren","doi":"10.1016/j.amc.2025.129934","DOIUrl":"10.1016/j.amc.2025.129934","url":null,"abstract":"<div><div>Optimal consensus of multi-agent systems under stochastic disturbances yields valuable insights into improving collaborative efficiency, resource utilization, and system robustness amidst uncertain environments. This study focuses on achieving optimal consensus in fractional-order multi-agent systems (FOMASs) with additive noises, an emerging research area with significant theoretical and practical implications, which deepens our understanding of system behavior and supports precise modeling and control strategy development. First, we introduce an innovative approach using a frequency distribution model to bridge fractional-order and integer-order systems, deriving sufficient conditions for Lyapunov stability. Second, we establish and address an optimal control problem tailored for stochastic Caputo fractional-order systems, laying the foundation for exploring the balance between control performance and robustness in the presence of stochastic perturbations and fractional-order dynamics. Additionally, we tackle optimal consensus control problems for stochastic FOMASs, providing sufficient conditions for achieving optimal mean square consensus through optimal control strategies and consensus conditions. The complexity of solution expressions is heightened by the presence of random variables. Finally, practical applicability is demonstrated by applying these findings to the distributed circuits and battery circuit equalization problem, thereby expanding the application potential of fractional-order systems in renewable energy.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129934"},"PeriodicalIF":3.4,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-03DOI: 10.1016/j.amc.2025.129929
Manuel Montes-y-Morales, Saylé Sigarreta, Hugo Cruz-Suárez
In this paper, we present a dynamic programming approach for identifying extremal polyomino chains with respect to degree–based indices. This approach provides an explicit recurrence, and a constructive algorithm that enable both the computation of an extremal polyomino chain in linear time with respect to the number of squares, and the enumeration of all maximal configurations. As a main application, we focus on the problem posed in 2016 by characterizing the polyomino chains that maximize the Augmented Zagreb Index (AZI) for any fixed number of squares. The results presented in this paper are aligned with previous contributions, and establish a constructive methodology for solving extremal problems in chemical graph theory. A link to the implementation code is provided in the last section.
{"title":"Maximum augmented Zagreb index on polyomino chains","authors":"Manuel Montes-y-Morales, Saylé Sigarreta, Hugo Cruz-Suárez","doi":"10.1016/j.amc.2025.129929","DOIUrl":"10.1016/j.amc.2025.129929","url":null,"abstract":"<div><div>In this paper, we present a dynamic programming approach for identifying extremal polyomino chains with respect to degree–based indices. This approach provides an explicit recurrence, and a constructive algorithm that enable both the computation of an extremal polyomino chain in linear time with respect to the number of squares, and the enumeration of all maximal configurations. As a main application, we focus on the problem posed in 2016 by characterizing the polyomino chains that maximize the Augmented Zagreb Index (<em>AZI</em>) for any fixed number of squares. The results presented in this paper are aligned with previous contributions, and establish a constructive methodology for solving extremal problems in chemical graph theory. A link to the implementation code is provided in the last section.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129929"},"PeriodicalIF":3.4,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-02DOI: 10.1016/j.amc.2025.129931
Qianfen Liao , Yangming Li , Weijun Liu
<div><div>For an odd prime <em>p</em> ≠ 3, the cyclic group <em>Z</em><sub>3<em>p</em></sub>≅<em>Z</em><sub>3</sub> × <em>Z<sub>p</sub></em>, where <span><math><mrow><msub><mi>Z</mi><mn>3</mn></msub><mo>=</mo><mrow><mo>〈</mo><msub><mi>w</mi><mn>1</mn></msub><mo>〉</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mi>Z</mi><mi>p</mi></msub><mo>=</mo><mrow><mo>〈</mo><msub><mi>w</mi><mn>2</mn></msub><mo>〉</mo></mrow></mrow></math></span>. Let <em>β</em><sub>1</sub> and <em>β</em><sub>2</sub> be involutory automorphisms of <em>Z</em><sub>3<em>p</em></sub> defined by <span><math><mrow><msup><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>β</mi><mn>1</mn></msub></msup><mo>=</mo><mrow><mo>(</mo><msubsup><mi>w</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>β</mi><mn>2</mn></msub></msup><mo>=</mo><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>w</mi><mn>2</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>. Then the elements of sets <span><math><mrow><msub><mstyle><mi>Δ</mi></mstyle><mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>{</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>s</mi><mn>2</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>s</mi><mn>3</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>}</mo></mrow><mo>∣</mo><msub><mi>s</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>Z</mi><mn>3</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>≠</mo><msub><mi>s</mi><mn>3</mn></msub><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>≠</mo><msub><mi>e</mi><mn>2</mn></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mstyle><mi>Δ</mi></mstyle><mn>2</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>{</mo><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msubsup><mi>w</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msubsup><mi>w</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msub><mi>t</mi><mn>2<
{"title":"Isomorphisms of the 4-valent generalized Cayley graphs of Z3p","authors":"Qianfen Liao , Yangming Li , Weijun Liu","doi":"10.1016/j.amc.2025.129931","DOIUrl":"10.1016/j.amc.2025.129931","url":null,"abstract":"<div><div>For an odd prime <em>p</em> ≠ 3, the cyclic group <em>Z</em><sub>3<em>p</em></sub>≅<em>Z</em><sub>3</sub> × <em>Z<sub>p</sub></em>, where <span><math><mrow><msub><mi>Z</mi><mn>3</mn></msub><mo>=</mo><mrow><mo>〈</mo><msub><mi>w</mi><mn>1</mn></msub><mo>〉</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mi>Z</mi><mi>p</mi></msub><mo>=</mo><mrow><mo>〈</mo><msub><mi>w</mi><mn>2</mn></msub><mo>〉</mo></mrow></mrow></math></span>. Let <em>β</em><sub>1</sub> and <em>β</em><sub>2</sub> be involutory automorphisms of <em>Z</em><sub>3<em>p</em></sub> defined by <span><math><mrow><msup><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>β</mi><mn>1</mn></msub></msup><mo>=</mo><mrow><mo>(</mo><msubsup><mi>w</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>β</mi><mn>2</mn></msub></msup><mo>=</mo><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>w</mi><mn>2</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>. Then the elements of sets <span><math><mrow><msub><mstyle><mi>Δ</mi></mstyle><mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>{</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>s</mi><mn>2</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>s</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>s</mi><mn>3</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>}</mo></mrow><mo>∣</mo><msub><mi>s</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>Z</mi><mn>3</mn></msub><mo>,</mo><msub><mi>s</mi><mn>2</mn></msub><mo>≠</mo><msub><mi>s</mi><mn>3</mn></msub><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mi>s</mi><mn>2</mn></msub><mo>,</mo><msub><mi>s</mi><mn>3</mn></msub><mo>≠</mo><msub><mi>e</mi><mn>2</mn></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mstyle><mi>Δ</mi></mstyle><mn>2</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>{</mo><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msubsup><mi>w</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msub><mi>w</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><msubsup><mi>w</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><msub><mi>t</mi><mn>2<","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"519 ","pages":"Article 129931"},"PeriodicalIF":3.4,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145886438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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