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15 pages, 5932 KiB  
Article
Numerical Simulation of Fluid Flow, Heat Transfer, and Solidification in AISI 304 Stainless Steel Twin-Roll Strip Casting
by Jingzhou Lu, Wanlin Wang and Kun Dou
Metals 2025, 15(7), 749; https://doi.org/10.3390/met15070749 - 2 Jul 2025
Viewed by 233
Abstract
The production of AISI 304 stainless steel (a corrosion-resistant alloy prone to solidification defects from high alloy content) particularly benefits from twin-roll strip casting—a short-process green technology enabling sub-rapid solidification (the maximum cooling rate exceeds 1000 °C/s) control for high-performance steels. However, the [...] Read more.
The production of AISI 304 stainless steel (a corrosion-resistant alloy prone to solidification defects from high alloy content) particularly benefits from twin-roll strip casting—a short-process green technology enabling sub-rapid solidification (the maximum cooling rate exceeds 1000 °C/s) control for high-performance steels. However, the internal phenomena within its molten pool remain exceptionally challenging to monitor. This study developed a multiscale numerical model to simulate coupled fluid flow, heat transfer, and solidification in AISI 304 stainless steel twin-roll strip casting. A quarter-symmetry 3D model captured macroscopic transport phenomena, while a slice model resolved mesoscopic solidification structure. Laboratory experiments had verified that the deviation between the predicted temperature field and the measured average value (1384.3 °C) was less than 5%, and the error between the solidification structure simulation and the electron backscatter diffraction (EBSD) data was within 5%. The flow field and flow trajectory showed obvious recirculation zones: the center area was mainly composed of large recirculation zones, and many small recirculation zones appeared at the edges. Parameter studies showed that, compared with the high superheat (110 °C), the low superheat (30 °C) increased the total solid fraction by 63% (from 8.3% to 13.6%) and increased the distance between the kiss point and the bottom of the molten pool by 154% (from 6.2 to 15.8 mm). The location of the kiss point is a key industrial indicator for assessing solidification integrity and the risk of strip fracture. In terms of mesoscopic solidification structure, low superheat promoted the formation of coarse columnar crystals (equiaxed crystals accounted for 8.9%), while high superheat promoted the formation of equiaxed nucleation (26.5%). The model can be used to assist in the setting of process parameters and process optimization for twin-roll strip casting. Full article
(This article belongs to the Special Issue Advances in Metal Rolling Processes)
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12 pages, 918 KiB  
Article
Fault-Tolerant Edge Metric Dimension of Zero-Divisor Graphs of Commutative Rings
by Omaima Alshanquiti, Malkesh Singh and Vijay Kumar Bhat
Axioms 2025, 14(7), 499; https://doi.org/10.3390/axioms14070499 - 26 Jun 2025
Viewed by 220
Abstract
In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring [...] Read more.
In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring S with unity. Consider a graph Γ(S) with vertex set V(Γ) = Z*(S), and two vertices in Γ(S) are adjacent if and only if their product is zero. This graph Γ(S) is known as zero-divisor graph of S. Zero-divisor graphs provide a powerful bridge between abstract algebra and graph theory. The zero-divisor graphs for finite commutative rings and their minimum fault-tolerant edge-resolving sets are studied in this article. Through analytical and constructive techniques, we highlight how the algebraic properties of the ring influence the edge metric structure of its associated graph. In addition to this, the existence of a connected graph G having a resolving set of cardinality of 2n + 2 from a star graph K1,2n, is studied. Full article
(This article belongs to the Special Issue Recent Developments in Graph Theory)
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33 pages, 3629 KiB  
Article
An Artificial Plant Community Algorithm for Collision-Free Multi-Robot Aggregation
by Zhengying Cai, Qingqing Yu, Zhuimeng Lu, Zeya Liu and Guoqiang Gong
Appl. Sci. 2025, 15(8), 4240; https://doi.org/10.3390/app15084240 - 11 Apr 2025
Viewed by 315
Abstract
Multi-robot aggregation is an important application for emergent robotic tasks, in which multiple robots are aggregated to work collaboratively. In this context, the collision-free problem poses a significant challenge, which is complicated to resolve, as aggregated robots are prone to collision. This study [...] Read more.
Multi-robot aggregation is an important application for emergent robotic tasks, in which multiple robots are aggregated to work collaboratively. In this context, the collision-free problem poses a significant challenge, which is complicated to resolve, as aggregated robots are prone to collision. This study attempts to use robotic edge intelligence technology to solve this problem. First, a multiple objective function is built for the collision-free multi-robot aggregation problem, considering the characteristics of robotic aggregation and collision-free constraints. Second, a heuristic artificial plant community algorithm is proposed to obtain an optimal solution to the developed problem model, which is lightweight and can be deployed on edge robots to search for the optimal route in real-time. The proposed algorithm utilizes the swarm learning capability of edge robots to produce a set of collision-free aggregation assignments for all robots. Finally, a benchmark test set is developed, based on which a series of benchmark tests are conducted. The experimental results demonstrate that the proposed method is effective and its computational performance is suitable for robot edge computing. Full article
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30 pages, 1916 KiB  
Article
Zero-Trust Medical Image Sharing: A Secure and Decentralized Approach Using Blockchain and the IPFS
by Ali Shahzad, Wenyu Chen, Yin Zhang and Rajesh Kumar
Symmetry 2025, 17(4), 551; https://doi.org/10.3390/sym17040551 - 3 Apr 2025
Viewed by 1320
Abstract
The secure and efficient storage and sharing of medical images have become increasingly important due to rising security threats and performance limitations in existing healthcare systems. Centralized systems struggle to provide adequate privacy, rapid access, and reliable storage for sensitive medical images. This [...] Read more.
The secure and efficient storage and sharing of medical images have become increasingly important due to rising security threats and performance limitations in existing healthcare systems. Centralized systems struggle to provide adequate privacy, rapid access, and reliable storage for sensitive medical images. This paper proposes a decentralized medical image-sharing framework to address these issues by integrating blockchain technology, the InterPlanetary File System (IPFS), and edge computing. Blockchain technology enforces secure patient-centric access control through smart contracts that enable patients to directly manage their data-sharing permissions. The IPFS provides decentralized and scalable storage for medical images and effectively resolves the storage limitations associated with blockchain. Edge computing enhances system responsiveness by significantly reducing latency through local data processing to ensure timely medical image access. Robust security is ensured by using elliptic curve cryptography (ECC) for secure key management and the Advanced Encryption Standard (AES) for encrypting medical images to protect against unauthorized access and data breaches. Additionally, the system includes real-time monitoring to promptly detect and respond to unauthorized access attempts to ensure continuous protection against potential security threats. System results demonstrate that the proposed framework achieves lower latency, higher throughput, and improved security compared to traditional centralized storage solutions, which makes our system suitable for practical deployment in modern healthcare settings. Full article
(This article belongs to the Section Computer)
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14 pages, 9481 KiB  
Article
The One-Fault Dimension-Balanced Hamiltonian Problem in Toroidal Mesh Graphs
by Justie Su-Tzu Juan, Hao-Cheng Ciou and Meng-Jyun Lin
Symmetry 2025, 17(1), 93; https://doi.org/10.3390/sym17010093 - 9 Jan 2025
Cited by 1 | Viewed by 628
Abstract
Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge [...] Read more.
Finding a Hamiltonian cycle in a graph G = (V, E) is a well-known problem. The challenge of finding a Hamiltonian cycle that avoids these faults when faulty vertices or edges are present has been extensively studied. When the edge set of G is partitioned into k dimensions, the problem of dimension-balanced Hamiltonian cycles arises, where the Hamiltonian cycle uses approximately the same number of edges from each dimension (differing by at most one). This paper studies whether a dimension-balanced Hamiltonian cycle (DBH) exists in toroidal mesh graphs Tm,n when a single vertex or edge is faulty, called the one-fault DBH problem. We establish that Tm,n is one-fault DBH, except in the following cases: (1) both m and n are even; (2) one of m and n is 3, while the other satisfies mod 4 = 3 and is greater than 6; (3) one of m and n is odd, while the other satisfies mod 4 = 2. Additionally, this paper resolves a conjecture from prior literature, thereby providing a complete solution to the DBP problem on Tm,n. Full article
(This article belongs to the Section Mathematics)
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19 pages, 9100 KiB  
Article
Deep Ultraviolet Excitation Photoluminescence Characteristics and Correlative Investigation of Al-Rich AlGaN Films on Sapphire
by Zhe Chuan Feng, Ming Tian, Xiong Zhang, Manika Tun Nafisa, Yao Liu, Jeffrey Yiin, Benjamin Klein and Ian Ferguson
Nanomaterials 2024, 14(21), 1769; https://doi.org/10.3390/nano14211769 - 4 Nov 2024
Viewed by 1397
Abstract
AlGaN is attractive for fabricating deep ultraviolet (DUV) optoelectronic and electronic devices of light-emitting diodes (LEDs), photodetectors, high-electron-mobility field-effect transistors (HEMTs), etc. We investigated the quality and optical properties of AlxGa1−xN films with high Al fractions (60–87%) grown on [...] Read more.
AlGaN is attractive for fabricating deep ultraviolet (DUV) optoelectronic and electronic devices of light-emitting diodes (LEDs), photodetectors, high-electron-mobility field-effect transistors (HEMTs), etc. We investigated the quality and optical properties of AlxGa1−xN films with high Al fractions (60–87%) grown on sapphire substrates, including AlN nucleation and buffer layers, by metal–organic chemical vapor deposition (MOCVD). They were initially investigated by high-resolution X-ray diffraction (HR-XRD) and Raman scattering (RS). A set of formulas was deduced to precisely determine x(Al) from HR-XRD data. Screw dislocation densities in AlGaN and AlN layers were deduced. DUV (266 nm) excitation RS clearly exhibits AlGaN Raman features far superior to visible RS. The simulation on the AlGaN longitudinal optical (LO) phonon modes determined the carrier concentrations in the AlGaN layers. The spatial correlation model (SCM) analyses on E2(high) modes examined the AlGaN and AlN layer properties. These high-x(Al) AlxGa1−xN films possess large energy gaps Eg in the range of 5.0–5.6 eV and are excited by a DUV 213 nm (5.8 eV) laser for room temperature (RT) photoluminescence (PL) and temperature-dependent photoluminescence (TDPL) studies. The obtained RTPL bands were deconvoluted with two Gaussian bands, indicating cross-bandgap emission, phonon replicas, and variation with x(Al). TDPL spectra at 20–300 K of Al0.87Ga0.13N exhibit the T-dependences of the band-edge luminescence near 5.6 eV and the phonon replicas. According to the Arrhenius fitting diagram of the TDPL spectra, the activation energy (19.6 meV) associated with the luminescence process is acquired. In addition, the combined PL and time-resolved photoluminescence (TRPL) spectroscopic system with DUV 213 nm pulse excitation was applied to measure a typical AlGaN multiple-quantum well (MQW). The RT TRPL decay spectra were obtained at four wavelengths and fitted by two exponentials with fast and slow decay times of ~0.2 ns and 1–2 ns, respectively. Comprehensive studies on these Al-rich AlGaN epi-films and a typical AlGaN MQW are achieved with unique and significant results, which are useful to researchers in the field. Full article
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20 pages, 5007 KiB  
Article
Multi-Material Topology Optimization on Separate Tetrahedral Meshes with Explicit Design Resolution by Means of Remeshing
by Robert Renz and Albert Albers
Algorithms 2024, 17(10), 460; https://doi.org/10.3390/a17100460 - 16 Oct 2024
Viewed by 1463
Abstract
As a method of lightweight design, multi-material design aims to make targeted use of materials in order to reduce CO2 emissions. In this context, it can be described as one of the product development methods used to meet the challenges of climate [...] Read more.
As a method of lightweight design, multi-material design aims to make targeted use of materials in order to reduce CO2 emissions. In this context, it can be described as one of the product development methods used to meet the challenges of climate change. However, since the design of structures in multi-material design is complex, topology optimization can be used to support the product developer. In this article, a multi-material topology optimization method is developed that combines the Velocity Field Level Set method with the Reconciled Level Set method. Furthermore, the current design is explicitly resolved in each iteration by means of multi-material remeshing. The edge collapse phase in the remeshing process is achieved by applying the producer consumer pattern. The developed method is then validated using known examples from the state of research, and the influence of the parameters of the method on the result is analyzed by means of studies. Full article
(This article belongs to the Section Analysis of Algorithms and Complexity Theory)
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7 pages, 1266 KiB  
Communication
Unusual Unsatisfactory Treatment in Two Patients with Imported Cutaneous Leishmaniasis
by Anna Kuna, Romuald Olszański, Beata Szostakowska, Natalia Kulawiak, Ravi Kant and Maciej Grzybek
Trop. Med. Infect. Dis. 2024, 9(10), 227; https://doi.org/10.3390/tropicalmed9100227 - 30 Sep 2024
Viewed by 2028
Abstract
Cutaneous leishmaniasis is one of the most commonly diagnosed dermatological condition in travel medicine after diarrhoeal diseases and febrile status. The disease is transmitted by Phlebotomus and Lutzomyia sandflies. It appears in various clinical forms, the most common of which is a painless [...] Read more.
Cutaneous leishmaniasis is one of the most commonly diagnosed dermatological condition in travel medicine after diarrhoeal diseases and febrile status. The disease is transmitted by Phlebotomus and Lutzomyia sandflies. It appears in various clinical forms, the most common of which is a painless ulcer with raised edges, usually present on exposed parts of the body on the side where the insect bite occurred. Annually, over a million new cutaneous leishmaniasis (CL) cases are reported globally. We present two cases of affliction, the first occurring in Patient 1, who attempted treatment through the Kambo cleanse in South America, which is considered a toxic, even life-threatening, procedure. It involves the subcutaneous application of a substance dangerous to humans derived from the surface mucus of a frog. Patient 2 applied caustic ointments, a fruitarian diet, and hyperbaric oxygen therapy in a private setting. After initial therapeutic failures caused by the patients’ unconventional treatment ideas, the causal treatment effect was satisfactory, demonstrating the efficacy of these treatments in resolving the infection when applied appropriately. Despite the typical CL presentation in both patients, their self-treatment course was unusual. It is worth noting that alternative, sometimes harmful, self-treatment initiatives by patients may be surprising and ineffective. Promoting knowledge about tropical diseases among travellers and medical staff is crucial to improving treatment outcomes. Full article
(This article belongs to the Section Neglected and Emerging Tropical Diseases)
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18 pages, 318 KiB  
Article
Structural Analysis of Octagonal Nanotubes via Double Edge-Resolving Partitions
by Amal S. Alali, Sikander Ali and Muhammad Kamran Jamil
Processes 2024, 12(9), 1920; https://doi.org/10.3390/pr12091920 - 6 Sep 2024
Cited by 3 | Viewed by 1525
Abstract
In materials science, the open nanotube derived from an octagonal grid is one of the most important and extensively researched compounds. Finding strategies for representing a variety of chemical compounds so that different compounds can have different representations is necessary for the investigation [...] Read more.
In materials science, the open nanotube derived from an octagonal grid is one of the most important and extensively researched compounds. Finding strategies for representing a variety of chemical compounds so that different compounds can have different representations is necessary for the investigation of chemical structures. In this work, the double edge-based resolving partition is discussed and the exchange property applied. Let Q1 and Q2 be two edge-resolving partition sets and Q1Q2, such that Q1Q20. This shows that this structure has exchange property for edge partition. The exchange property in edge partitions is a novel work. It is introduced in this paper. The application of this work is to transform projects or objects to better places. The resolvability of these compounds is studied to gain an understanding of the chemical composition of the compounds. We perform this by using the terms vertex and edge-based distance and edge-resolving sets of graphs. Full article
(This article belongs to the Section Chemical Processes and Systems)
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19 pages, 4829 KiB  
Article
Robust Frontal Vehicle Pose Estimation Based on Structural Parameter Optimization Using Reliable Edge Point Sequences
by Jiang Chen, Weiwei Zhang, Miao Liu, Xiaolan Wang and Hong Li
Appl. Sci. 2023, 13(24), 12993; https://doi.org/10.3390/app132412993 - 5 Dec 2023
Cited by 1 | Viewed by 1304
Abstract
In order to enhance the stability of vehicle pose estimation within driving videos, a novel methodology for optimizing vehicle structural parameters is introduced. This approach hinges on evaluating the reliability of edge point sequences. Firstly, a multi−task and iterative convolutional neural network (MI−CNN) [...] Read more.
In order to enhance the stability of vehicle pose estimation within driving videos, a novel methodology for optimizing vehicle structural parameters is introduced. This approach hinges on evaluating the reliability of edge point sequences. Firstly, a multi−task and iterative convolutional neural network (MI−CNN) is constructed, enabling the simultaneous execution of four critical tasks: vehicle detection, yaw angle prediction, edge point location, and visibility assessment. Secondly, an imperative aspect of the methodology involves establishing a local tracking search area. This region is determined by modeling the limitations of vehicle displacement between successive frames. Vehicles are matched using a maximization approach that leverages point similarity. Finally, a reliable edge point sequence plays a pivotal role in resolving structural parameters robustly. The Gaussian mixture distribution of vehicle distance change ratios, derived from two measurement models, is employed to ascertain the reliability of the edge point sequence. The experimental results showed that the mean Average Precision (mAP) achieved by the MI−CNN network stands at 89.9%. A noteworthy observation is that the proportion of estimated parameters whose errors fall below the threshold of 0.8 m consistently surpasses the 85% mark. When the error threshold is set at less than 0.12 m, the proportion of estimated parameters meeting this criterion consistently exceeds 90%. Therefore, the proposed method has better application status and estimation precision. Full article
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31 pages, 7030 KiB  
Article
OptiDJS+: A Next-Generation Enhanced Dynamic Johnson Sequencing Algorithm for Efficient Resource Scheduling in Distributed Overloading within Cloud Computing Environment
by Pallab Banerjee, Sharmistha Roy, Umar Muhammad Modibbo, Saroj Kumar Pandey, Parul Chaudhary, Anurag Sinha and Narendra Kumar Singh
Electronics 2023, 12(19), 4123; https://doi.org/10.3390/electronics12194123 - 2 Oct 2023
Cited by 8 | Viewed by 2898
Abstract
The continuously evolving world of cloud computing presents new challenges in resource allocation as dispersed systems struggle with overloaded conditions. In this regard, we introduce OptiDJS+, a cutting-edge enhanced dynamic Johnson sequencing algorithm made to successfully handle resource scheduling challenges in cloud computing [...] Read more.
The continuously evolving world of cloud computing presents new challenges in resource allocation as dispersed systems struggle with overloaded conditions. In this regard, we introduce OptiDJS+, a cutting-edge enhanced dynamic Johnson sequencing algorithm made to successfully handle resource scheduling challenges in cloud computing settings. With a solid foundation in the dynamic Johnson sequencing algorithm, OptiDJS+ builds upon it to suit the demands of modern cloud infrastructures. OptiDJS+ makes use of sophisticated optimization algorithms, heuristic approaches, and adaptive mechanisms to improve resource allocation, workload distribution, and task scheduling. To obtain the best performance, this strategy uses historical data, dynamic resource reconfiguration, and adaptation to changing workloads. It accomplishes this by utilizing real-time monitoring and machine learning. It takes factors like load balance and make-up into account. We outline the design philosophies, implementation specifics, and empirical assessments of OptiDJS+ in this work. Through rigorous testing and benchmarking against cutting-edge scheduling algorithms, we show the better performance and resilience of OptiDJS+ in terms of reaction times, resource utilization, and scalability. The outcomes underline its success in reducing resource contention and raising service quality generally in cloud computing environments. In contexts where there is distributed overloading, OptiDJS+ offers a significant advancement in the search for effective resource scheduling solutions. Its versatility, optimization skills, and improved decision-making procedures make it a viable tool for tackling the resource allocation issues that cloud service providers and consumers encounter daily. We think that OptiDJS+ opens the way for more dependable and effective cloud computing ecosystems, assisting in the full realization of cloud technologies’ promises across a range of application areas. In order to use the OptiDJS+ Johnson sequencing algorithm for cloud computing task scheduling, we provide a two-step procedure. After examining the links between the jobs, we generate a Gantt chart. The Gantt chart graph is then changed into a two-machine OptiDJS+ Johnson sequencing problem by assigning tasks to servers. The OptiDJS+ dynamic Johnson sequencing approach is then used to minimize the time span and find the best sequence of operations on each server. Through extensive simulations and testing, we evaluate the performance of our proposed OptiDJS+ dynamic Johnson sequencing approach with two servers to that of current scheduling techniques. The results demonstrate that our technique greatly improves performance in terms of makespan reduction and resource utilization. The recommended approach also demonstrates its ability to scale and is effective at resolving challenging work scheduling problems in cloud computing environments. Full article
(This article belongs to the Special Issue Empowering Sensor Applications with AI and Big Data Analytics)
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15 pages, 2055 KiB  
Article
Electromigration-Aware Memory Hierarchy Architecture
by Freddy Gabbay and Avi Mendelson
J. Low Power Electron. Appl. 2023, 13(3), 44; https://doi.org/10.3390/jlpea13030044 - 11 Jul 2023
Cited by 3 | Viewed by 2102
Abstract
New mission-critical applications, such as autonomous vehicles and life-support systems, set a high bar for the reliability of modern microprocessors that operate in highly challenging conditions. However, while cutting-edge integrated circuit (IC) technologies have intensified microprocessors by providing remarkable reductions in the silicon [...] Read more.
New mission-critical applications, such as autonomous vehicles and life-support systems, set a high bar for the reliability of modern microprocessors that operate in highly challenging conditions. However, while cutting-edge integrated circuit (IC) technologies have intensified microprocessors by providing remarkable reductions in the silicon area and power consumption, they also introduce new reliability challenges through the complex design rules they impose, creating a significant hurdle in the design process. In this paper, we focus on electromigration (EM), which is a crucial factor impacting IC reliability. EM refers to the degradation process of IC metal nets when used for both power supply and interconnecting signals. Typically, EM concerns have been addressed at the backend, circuit, and layout levels, where EM rules are enforced assuming extreme conditions to identify and resolve violations. This study presents new techniques that leverage architectural features to mitigate the effect of EM on the memory hierarchy of modern microprocessors. Architectural approaches can reduce the complexity of solving EM-related violations, and they can also complement and enhance common existing methods. In this study, we present a comprehensive simulation analysis that demonstrates how the proposed solution can significantly extend the lifetime of a microprocessor’s memory hierarchy with minimal overhead in terms of performance, power, and area while relaxing EM design efforts. Full article
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9 pages, 242 KiB  
Article
Resolvability in Subdivision Graph of Circulant Graphs
by Syed Ahtsham Ul Haq Bokhary, Khola Wahid, Usman Ali, Shreefa O. Hilali, Mohammed Alhagyan and Ameni Gargouri
Symmetry 2023, 15(4), 867; https://doi.org/10.3390/sym15040867 - 5 Apr 2023
Cited by 3 | Viewed by 1879
Abstract
Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts [...] Read more.
Circulant networks are a very important and widely studied class of graphs due to their interesting and diverse applications in networking, facility location problems, and their symmetric properties. The structure of the graph ensures that it is symmetric about any line that cuts the graph into two equal parts. Due to this symmetric behavior, the resolvability of these graph becomes interning. Subdividing an edge means inserting a new vertex on the edge that divides it into two edges. The subdivision graph G is a graph formed by a series of edge subdivisions. In a graph, a resolving set is a set that uniquely identifies each vertex of the graph by its distance from the other vertices. A metric basis is a resolving set of minimum cardinality, and the number of elements in the metric basis is referred to as the metric dimension. This paper determines the minimum resolving set for the graphs Hl[1,k] constructed from the circulant graph Cl[1,k] by subdividing its edges. We also proved that, for k=2,3, this graph class has a constant metric dimension. Full article
(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)
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26 pages, 7620 KiB  
Article
A New Technique to Uniquely Identify the Edges of a Graph
by Hafiz Muhammad Ikhlaq, Rashad Ismail, Hafiz Muhammad Afzal Siddiqui and Muhammad Faisal Nadeem
Symmetry 2023, 15(3), 762; https://doi.org/10.3390/sym15030762 - 20 Mar 2023
Cited by 5 | Viewed by 2554
Abstract
Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the [...] Read more.
Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, BV(Γ). If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions. Full article
(This article belongs to the Special Issue Theoretical Computer Science and Discrete Mathematics II)
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14 pages, 341 KiB  
Article
Constant Time Calculation of the Metric Dimension of the Join of Path Graphs
by Chuanjun Zhang, Ghulam Haidar, Murad Ul Islam Khan, Faisal Yousafzai, Kostaq Hila and Asad Ul Islam Khan
Symmetry 2023, 15(3), 708; https://doi.org/10.3390/sym15030708 - 12 Mar 2023
Cited by 5 | Viewed by 1806
Abstract
The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset [...] Read more.
The distance between two vertices of a simple connected graph G, denoted as d(u,v), is the length of the shortest path from u to v and is always symmetrical. An ordered subset W=w1,w2,w3,,wk of V(G) is a resolving set for G, if for ∀u,vV(G), there exists wiWd(u,wi)d(v,wi). A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as dim(G). For the graph G1=(V1,E1,) and G2=(V2,E2), their join is denoted by G1+G2. The vertex set of G1+G2 is V1V2 and the edge set is E=E1E2uv,uV1,vV2. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries. Full article
(This article belongs to the Section Mathematics)
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