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                <text>Joseph, Ancy Kandathil; Kureethara, Joseph Varghese</text>
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                <text>Antimagic labeling of n-uniform cactus chain graphs</text>
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                <text>Discrete Mathematics, Algorithms and Applications;Volume;18;Issue;1;Article No.;2550015;</text>
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                <text>Joseph A.K., Department of Mathematics, Christ University, Bangalore, 560029, India; Kureethara J.V., Department of Mathematics, Christ University, Bangalore, 560029, India</text>
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                <text>A graph G = (V,E) is considered antimagic if it admits antimagic labeling. The antimagic labeling of a finite, simple graph with |V | = n and |E| = m is a bijective function from the set of edges to the set of integers {1, 2,,m} such that the vertex sum of n vertices is pairwise distinct. The vertex sum of a vertex is obtained by summing the labels of all edges incident to it. Hartsfield and Ringel conjectured that every connected graph different from K2 is antimagic. Supporting this conjecture, it was shown that the dense graphs are antimagic. A cactus graph is a connected graph where no edge lies within more than one cycle. A cactus graph in which each block is a cycle of the same size n is called an n-uniform cactus graph. We proved that Hartsfield and Ringels conjecture is true for n-uniform cactus chain graphs with and without pendant vertices, which are specific cases of sparse graphs.  2026 World Scientific Publishing Company.</text>
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                <text>antimagic labeling; cactus chain graph; equivalence class; Graph labeling</text>
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                <text>ISSN: 17938309;</text>
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                <text>Joseph, Anjali Merin; Prakash, L.T. Om</text>
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                <text>Technopedagogy in teacher education: exploring challenges and possibilities</text>
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                <text>International Journal of Knowledge and Learning;Volume;18;Issue;6;pp.549-564</text>
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                <text>&lt;a href="https://doi.org/10.1504/IJKL.2025.149853" target="_blank" rel="noreferrer noopener"&gt;https://doi.org/10.1504/IJKL.2025.149853&lt;/a&gt; &lt;br /&gt;&lt;br /&gt;&lt;a href="https://www.scopus.com/pages/publications/105021874659?origin=resultslist" target="_blank" rel="noreferrer noopener"&gt;https://www.scopus.com/pages/publications/105021874659?origin=resultslist&lt;/a&gt;</text>
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                <text>Joseph A.M., Department of Sociology, Christ University, Bangalore, India; Prakash L.T.O., Department of Sociology, Christ University, Bangalore, India</text>
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                <text>Digital technologies allowed teachers to overcome spatial and temporal limitations in education, particularly during the COVID-19 pandemic-imposed restrictions. While access to technological resources proved beneficial, teachers faced initial challenges. It is crucial to address the significance of digital education training in teacher education institutions, particularly in implementing the Integrated Teacher Education Programme based on the National Education Policy 2020 in India. This study explores approaches to techno-pedagogical skills in teacher education in Kerala, India, and the potential solutions to bridge the digital gap between training and teaching in the classroom. The researchers have used qualitative methods to gather and analyse data, including archival research and interviews with teacher educators and student teachers in the Bachelor of Education Programme. The findings indicate an urgent need for infrastructural upgrades and continuous professional development practices. Copyright  2025 Inderscience Enterprises Ltd.</text>
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                <text>digital education; teacher training; teaching-learning practices; technology</text>
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                <text>ISSN: 17411009;</text>
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                <text>Joseph, Ankita; George, Jossy; Alapatt, Bosco Paul</text>
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                <text>Breast Cancer Prediction using a Stacked Ensemble of XGBoost and LightGBM with Logistic Regression Meta-Learning</text>
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                <text>Proceedings of the 2026 International Conference on AI-Driven Smart Systems and Ubiquitous Computing, ICAUC 2026;pp.1366-1373</text>
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                <text>&lt;a href="https://doi.org/10.1109/ICAUC68182.2026.11440972" target="_blank" rel="noreferrer noopener"&gt;https://doi.org/10.1109/ICAUC68182.2026.11440972&lt;/a&gt; &lt;br /&gt;&lt;br /&gt;&lt;a href="https://www.scopus.com/pages/publications/105037460270?origin=resultslist" target="_blank" rel="noreferrer noopener"&gt;https://www.scopus.com/pages/publications/105037460270?origin=resultslist&lt;/a&gt;</text>
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                <text>Joseph A., Christ (Deemed to be University), Department of Computer Science, India; George J., Christ (Deemed to be University), Department of Computer Science, India; Alapatt B.P., Christ (Deemed to be University), Department of Computer Science, India</text>
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                <text>Breast cancer remains one of the major reasons for cancer deaths in women, which is why it is key to develop and improve diagnostic systems for accurate predictions. Currently, the advent of Machine learning has helped in providing powerful algorithms to achieve advancements in cancer detection. However, the main motivation of this research is to focus on building more complex ensemble architectures, as they are known for significantly improving predictive accuracy, robustness, and generalisation, especially in performing complex tasks such as medical diagnosis. In this research, a Hybrid stacking ensemble was built using two gradient boosting techniques, XGBoost and LightGBM, with a Logistic Regression meta-learner to predict breast cancer and compare their performance with standard classifiers. The Breast Cancer Wisconsin (Diagnostic) dataset, which consists of 569 patient records, was utilised for model training and analysis. The data was preprocessed using Z-score normalisation and stratified 5-fold cross-validation. The machine learning algorithms, such as Decision Tree, Logistic Regression, and Random Forest, were compared with the hybrid model, and the metrics used for comparison were accuracy, precision, recall, F1-score, and ROC-AUC. The proposed hybrid model performed well, achieving a high accuracy rate of 97.37% and a recall rate of 93.00% for malignant cases. McNemar's test (p &amp;gt; 0.05) confirms that this accuracy rate is statistically equivalent to the Random Forest classifier. These findings proved that the proposed model can perform optimally in predicting complex data with the same degree of precision as the standard models. Therefore, the hybrid model can be considered a robust and reliable new alternative for breast cancer prediction.   2026 IEEE.</text>
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                <text>Breast Cancer Prediction; Classifier; Hybrid Ensemble; LightGBM; Random Forest; Stacking; XGBoost</text>
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                <text>Institute of Electrical and Electronics Engineers Inc.</text>
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                <text>ISBN: 979-833155851-2;</text>
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                <text>On the Maximization of Some Graph Coloring Problems  </text>
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                <text>Mathematics and Statistics</text>
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                <text>A graph coloring problem involves labeling the vertices or edges in a graph with newlinecolors or numbers subject to some constraints. The most frequently known graph newlinecoloring problems are the ones that usually minimize the number of colors used in newlinecoloring the vertices or edges. The chromatic number of a graph G, denoted by and#967;(G), is the least number of colors used in a proper coloring of G. The chromatic sum of a graph G, denoted as P(G), was introduced in [1], which is to and the smallest possible coloring sum in a proper coloring of the graph G using natural numbers. Lately, a few studies have endured in a distinct area of the literature where the number of colors used in a graph coloring problem is maximized under certain conditions. Some of these works have applications in network sciences. newlineThe concerned study focuses on the maximization of three dierent edge coloring newlineconcepts, viz., the vertex induced kand#8722;edge coloring, vertex incident kand#8722;edge coloring, newlineand edge incident 2and#8722;edge coloring of a simple connected graph G, where k and#8805; 2. The newlinenumber of colors assigned to the edges of the graph G has been maximized under certain conditions. The vertex induced kand#8722;edge coloring and the vertex incident newlinekand#8722;edge coloring concepts are the generalized version of the edge coloring approach newlineintroduced and studied in [2]. Furthermore, the concept of the achromatic sum of a graph G has also been introduced here. This concept is to and the greatest possible coloring sum of the graph G in an improper edge coloring using natural numbers. An extensive study newlineon three achromatic sums, namely the vertex induced 2and#8722;edge coloring sum, the vertex incident 2and#8722;edge coloring sum, and the edge incident 2and#8722;edge coloring sum are carried out. A few bounds for these parameters on a simple connected graph G and the exact values for some elementary graph classes have been investigated. A few comparative results between some of these parameters have also been obtained.</text>
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                <text>Joseph, Anu</text>
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                <text>Joseph A., Department of Mathematics, CHRIST (Deemed to be University), Karnataka, Bangalore, 560029, India; Dominic C., Department of Mathematics, CHRIST (Deemed to be University), Karnataka, Bangalore, 560029, India</text>
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                <text>The edge incident 2-edge coloring of a graph G is an edge coloring of the graph G such that not more than two colors are assigned to the edges incident to an edge e = uv in G. In other words, for every edge e in G, the edge e and all the edges that are incident to the edge e is in at most two different color classes. The edge incident 2-edge coloring number ?'ein2 (G) is the maximum number of colors in any edge incident 2-edge coloring of G. The main objective of this paper is to study the edge incident 2-edge coloring concept and apply the same to some graph classes. Besides finding the exact values of these parameters, we also obtain some bounds.  2025 World Scientific Publishing Company.</text>
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                <text>Joseph A., New Horizon College of Engineering, Bangalore, 560103, India, James Cook University, Townsville, Australia; George S., Christ University, Bangalore, 560029, India, James Cook University, Townsville, Australia; Shatte A., James Cook University, Townsville, Australia</text>
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                <text>Quantum computing introduces a new paradigm in digital forensics by enabling faster cryptographic analysis, enhanced machine learning, and secure data acquisition. This research examines the potential to apply quantum computing to forensics and how it can be used to transform the field through its disruptive capabilities in the evidence collection process, detection of intrusions, modeling of cybercrime, and DNA analysis. It also underrates the dangers that quantum technologies bring to the data security and the urgency of post-quantum encryption technologies. The article presents a blueprint of quantum-driven forensic investigation of the near future by conducting a survey of recent advances and new applications. We combine theory and practice by using datasets such as NSL-KDD, Qiskit simulations, and diagrams of how quantum machine learning models, DNA profiling and intrusion detection systems are used. Pattern matching in the DNA profiling algorithm with quantum computing is determined to have a time complexity of O(n) in the application of the Grover algorithm and O(n) of the corresponding classical algorithm.  The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.</text>
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                <text>Joseph D., Christ University, Bangalore, India; James B.V., Christ University, Bangalore, India; Sharma T., Christ University, Bangalore, India</text>
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                <text>This study explores the complex dynamics between perception, psychological ownership, and re-subscription intentions among Gen Z users of OTT platforms, specifically examining how perceived benefits and perceived drawbacks shape user behavior and investigating the moderating role of psychological ownership in this context. The research focuses on a sample of Gen Z users from India who actively engage with OTT platforms, and data were collected through a structured questionnaire comprising three sections; a structural equation modeling (SEM) technique was applied to analyze the data obtained from 304 valid responses. The analysis reveals that perceived benefits significantly enhance Gen Z's resubscription intentions, while perceived drawbacks have a negative impact; moreover, the study highlights that psychological ownership moderates the influence of perceived drawbacks, mitigating their adverse effect on resubscription intentions. Although the study is limited to Gen Z users in India and focuses on a specific set of independent constructs, future research could expand this scope by incorporating other generational cohorts and a broader range of influencing factors to deepen the understanding of user behavior in diverse contexts. This research contributes to the broader literature on consumer behavior in the digital landscape by modeling the interaction between psychological and perceptual factors within a complex system, providing empirical evidence on the moderating role of psychological ownership and emphasizing the importance of these dynamics in designing effective engagement strategies for OTT platforms. Insights from this study underscore the significance of enhancing user perception factors to boost re-subscription rates, and industry practitioners are encouraged to focus on delivering personalized and memorable digital experiences to strengthen psychological ownership and minimize perceived drawbacks. The study also highlights practical strategies for OTT platforms, such as developing high-quality content, intuitive interfaces, and fostering a sense of community and ownership among users, with a focus on addressing perceived drawbacks and enhancing the social value of these platforms as crucial measures for retaining Gen Z users. As one of the first studies to employ complex systems modeling techniques to understand the interplay between perception factors and psychological ownership in influencing re-subscription intentions among Gen Z OTT users, the findings offer valuable insights for the online service industry to refine their service delivery and user engagement strategies.  2025, Binghamton University Libraries. All rights reserved.</text>
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                <text>This increased reliance on networks has made the security of IoT devices a highly burning issue. Among the sea of threats, the threats associated with DDoS attacks are at a high level since they do damage to the functionality of devices and unavailability of networks. Detection and mitigation of DDoS attacks will demand IoT environments based on powerful classification algorithms. This paper evaluates the performances of three prominent algorithms: Decision Tree, Random Forest, and Histogram-Based Gradient Boosting for the classification of DDoS attack traffic within IoT networks. An IoT-23 dataset comprising a subset of attacks, including DDoS, is used herein for the purpose of achieving high classification accuracy to ensure a reliable evaluation of attacks. The results clearly show that all three algorithms are pretty good in terms of detection performance, and Histogram-Based Gradient Boosting is the best in terms of generalization accuracy. These results open new perspectives for the implementation of machine learning, generally, and Histogram-Based Gradient Boosting, specifically, directed to improving security in IoT networks against DDoS attacks, which is an extremely promising result when working within the light of some insights for future research and development within this critical area of security.  The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2026.</text>
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                <text>In an era defined by volatility, uncertainty, complexity, and ambiguity (VUCA), organisational change is no longer episodicit is a continuous necessity. This chapter explores how organisations can navigate constant change in a volatile, uncertain, complex, and ambiguous (VUCA) world. It examines key drivers of change technological, demographic, environmental, and socio- political and integrates classical and contemporary theories, including Lewins model, Kotters framework, and sensemaking. Emphasis is placed on strategic foresight, organisational agility, innovation culture and role of neuroplasticity in change implementation. Real- world case studies (e.g., Microsoft, Patagonia, GM) demonstrate practical applications from organisational and neuroplasticity perspective. The chapter highlights the importance of adaptive resilience, aligning formal and informal systems, empowering middle managers, and fostering psychological safety to enable sustainable transformation and continuous learning in rapidly evolving environments.  2026 by IGI Global Scientific Publishing.</text>
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                <text>Is connection the key? The mediating role of psychological safety in the relationship between relatedness to employee engagement</text>
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                <text>This study examines the influence of Relatedness Needs (RL) on Employee Engagement (EE) in Bengaluru's Information Technology (IT) Industry, with the mediating role of psychological safety (PS). As the information technology industry experiences continuous innovation and is associated with a high-pressure work environment, aligning the organizational needs with the employee's needs is critical to ensure organizational success. Employees with higher RL needs satisfaction will exhibit positive commitment and higher engagement, contributing to the long-term productivity and success of the organisation. This study examines the extent to which PS, indicating a safe environment promoting transparent communication, sharing ideas and engaging in collaborative decision-making without the fear of negative consequences, mediates the relationship between RL and EE. To test the study's hypothesis, AMOS, Smart-PLS, and structural equation modeling were used to analyse data collected from 304 employees working across companies in Bengaluru's Information Technology (IT) industry. Our findings suggest that having a stronger RL boosts EE through the mediating role of PS characterised by trustworthiness, a sense of safety and fairness at the workplace. The results suggest that fostering higher RL and ensuring a strong PS is vital for sustained EE and reducing turnover intention. This study offers valuable insights into the Information technology (IT) companies intending to boost workforce engagement in a highly pressured work environment.  2025 The Authors</text>
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                <text>Joseph E.R., Christ University, India; Seshadri V., Christ University, India</text>
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                <text>This chapter undertakes a comprehensive exploration of the specific organisational characteristics that are foundational to the cultivation and sustained presence of happiness and well- being within the workplace. Moving beyond the well- trodden path of individual- level determinants of employee satisfaction, this contribution meticulously dissects the systemic elements, deeply ingrained cultural norms, and thoughtfully designed structural frameworks that collectively contribute to the emergence of a positive and flourishing work environment. By examining existing scholarly research and synthesizing diverse theoretical perspectives, this chapter identifies the core organisational attributes that not only nurture employee happiness and a profound sense of well- being but also exert a demonstrably positive influence on critical organisational outcomes.  2026 by IGI Global Scientific Publishing.</text>
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                <text>Advances in Developing Human Resources;Volume;28;Issue;2;pp.103-123</text>
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                <text>Joseph E.R., School of Business and Management, CHRIST (Deemed to be University), Bangalore, India; Seshadri V., School of Business and Management, CHRIST (Deemed to be University), Bangalore, India; Kumar A.A., School of Business and Management, CHRIST (Deemed to be University), Bangalore, India; Thankachan J., School of Business and Management, CHRIST (Deemed to be University), Bangalore, India</text>
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                <text>Problem: Widespread disengagement among the working professionals and changing career expectations of Generation Z and Millennials pose a risk to talent management, retention, and performance in the IT sector. Solution: This quantitative study examines how employees perceived competence translates into engagement via multi-directional career development pathways (career lattice, career-goal progress, professional ability improvement, and promotion) using a survey of 304 IT professionals and PLS-SEM. Results show that employees who felt more competent reported higher engagement, and this relationship was strengthened when organisations provided career lattice pathways such as skill development, lateral mobility, and goal progress. Stakeholders: The findings identify actionable levers for organizational leaders, HRD practitioners, and talent managers to design adaptive career frameworks that increase engagement among Gen Z and Millennial employees. Implication: HRD leaders and talent managers should prioritise flexible, skills-based mobility and career-goal supports to boost Gen Z and Millennials engagement.  The Author(s) 2026</text>
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                <text>Crossroads;Volume;2025;Issue;50;pp.52-68</text>
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                <text>Joseph G., CHRIST University, Karnataka, Bangalore, India; Kennedy J.J., CHRIST University, Karnataka, Bangalore, India</text>
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                <text>Indigenous writing with postcolonial themes foregrounds the erasure and marginalization that result from colonialism. The genre-disrupting, coming-of-age novel Split Tooth (2018) by Inuit author Tanya Tagaq explores the personal and public life of a young Inuk woman from one of the Indigenous communities in the Canadian Arctic region. Split Tooth focuses on themes like the disappearances and deaths of Indigenous women, Inuit cultural settings, sexual assault, precarity, and violence. The novel meanders through emotions such as fear, shame, and grief, and can be analyzed through the theoretical framework of postcolonial affect. Postcolonial affect primarily examines the diverse emotional states of the colonized as indicators of the crisis that arises from colonization. The objective of the analysis is to highlight the delineation of affect in Split Tooth, as Tagaq blends the personal and the political in her narrative. Postcolonial affect is used for the theoretical examination of appropriation and violence that constitute the precarity of Inuit people, particularly women.  2025, Faculty of Philology, University of Bialystok. All rights reserved.</text>
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                <text>Contemporary higher education emphasizes glocal frameworks that accommodate both global and regional priorities in development. Decolonization of education becomes crucial in this context, where colonization led to the marginalization of Indigenous cultures, including Native American cultural traditions and knowledge systems. Multicultural inclusivity in higher education observes marginalized epistemic systems and gives them a place in the curriculum. By incorporating Native American texts in the English studies university syllabi in India, decolonization is brought into purview. This chapter deals with how the South Indian universities introduce Native American texts. The authors argue that multiculturalism in this context contributes to decolonial pedagogy, and the existing epistemic gap due to colonial oppression is problematized by inspecting the epistemic traditions of former colonies outside India. Content analysis of syllabi in the Central and state universities is the qualitative method used.  2025 by IGI Global Scientific Publishing. All rights reserved.</text>
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                <text>Lecture Notes in Networks and Systems;Volume;1354 LNNS;pp.465-476</text>
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                <text>&lt;a href="https://doi.org/10.1007/978-981-96-4880-1_37" target="_blank" rel="noreferrer noopener"&gt;https://doi.org/10.1007/978-981-96-4880-1_37&lt;/a&gt; &lt;br /&gt;&lt;br /&gt;&lt;a href="https://www.scopus.com/pages/publications/105019317616?origin=resultslist" target="_blank" rel="noreferrer noopener"&gt;https://www.scopus.com/pages/publications/105019317616?origin=resultslist&lt;/a&gt;</text>
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                <text>Joseph H., Department of Computer Science and Engineering, School of Engineering and Technology, Christ University, Karnataka, Bangalore, India; Manjus E., Department of Computer Science and Engineering, School of Engineering and Technology, Christ University, Karnataka, Bangalore, India; Arun Kokatnoor S., Department of Computer Science and Engineering, School of Engineering and Technology, Christ University, Karnataka, Bangalore, India; Bindu Madavi K.P., Department of Computer Science and Engineering, School of Engineering and Technology, Christ University, Karnataka, Bangalore, India</text>
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                <text>Malicious software must be detected in order to protect sensitive data and systems in the digital era, as sophisticated malware is posing serious risks to cybersecurity. By examining supervised machine learning approaches with a particular focus on Random Forest, Logistic Regression, and Decision Trees, this research proposes a data-driven approach to malware detection. These algorithms are trained to recognize patterns indicating malware by using labeled datasets containing four types of malwares, Ransomware, Trojan, Virus, and Worm. The performance of these algorithms is comprehensively investigated in the paper, with comparisons made between their accuracy, precision, recall, and F1-score. Based on the experimental results, Random Forest (96% accuracy) performed better in terms of robustness and accuracy of detection than both Logistic Regression (91%) and Decision Trees (84%). Logistic Regression provided faster computation at the expense of less accurate detection. Decision trees, while relatively simple to comprehend, performed moderately and they overfit the data. The studys conclusion highlights the significance of choosing the appropriate model in accordance with particular cyber security requirements, outlining the advantages and disadvantages of every approach as well as their practical applicability in real-time malware detection systems.  The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.</text>
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                <text>Decision trees and supervised; Logistic regression; Machine learning; Malware detection; Random forest</text>
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                <text>STRUCTURAL PROPERTIES OF SIGNED GRAPHS ADMITTING ROMAN DOMINATING FUNCTION</text>
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                <text>Turkish World Mathematical Society Journal of Applied and Engineering Mathematics;Volume;15;Issue;2;pp.353-362</text>
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                <text>Joseph J., Department of Mathematics, CHRIST(Deemed to be University), Bangalore, India; Joseph M., Department of Mathematics, CHRIST(Deemed to be University), Bangalore, India</text>
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                <text>A Roman dominating function(RDF) on a signed graph S = (G, ?) is a function f: V (S) ? {0, 1, 2} such that (i) (Formula presented) for every vertex (Formula presented) and (ii) for any vertex v with f(v) = 0 there exists a vertex (Formula presented) having f(u) = 2. In this article we explore structural properties of signed graphs admitting an RDF. Further, signed graphs with 3-regular graph as their underlying graph are examined and characterisation of one of its subclasses, net-regular signed graphs admitting an RDF is obtained.  I??k University, Department of Mathematics, 2025; all rights reserved.</text>
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                <text>roman dominating function; roman domination number; signed graphs</text>
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                <text>A study on dominating functions in signed graphs </text>
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                <text>Joseph, James&#13;
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                <text>Christ(Deemed to be University)&#13;
&#13;
&#13;
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                <text>2024-11-14</text>
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                <text>English&#13;
&#13;
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                <text>Joseph, Mayamma&#13;
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                <text>In this thesis, a study on Roman dominating functions in the realm of signed graphs is carried out. Unlike graphs, not all signed graphs admit a Roman dominating function, which leads to the primary problem of exploring signed graphs admitting a Roman dominating function. Further, variants of dominating function such as Roman {2}-dominating function, Minus dominating function and Signed dominating function in signed graphs are also studied. A dominating set of a signed graph S is dened as a set D and#8838; V such that each vertex v and#8712; V \ D has at least one neighbour u and#8712; D with and#963;(uv) = 1. The domination number and#947;(S) is the minimum cardinality among all the dominating sets newlineof S. A characterisation for minimal dominating sets of a signed graph along with newlinecharacterisations of signed graphs with domination number k, where 1 and#8804; k and#8804; 4 and newlinen and#8722; 2 and#8804; k and#8804; n are obtained. A Roman dominating function(RDF) on a signed graph S = (G, and#963;) is a function f : V (S) and#8594; {0, 1, 2} having the properties that (i) for every vertex u and#8712; V (G), f(N[u]) = f(u) + Pvand#8712;N (u) and#963;(uv)f(v) and#8805; 1 and (ii) for each vertex u and#8712; V (G) with f(u) = 0, there exists a vertex v and#8712; N +(u) having f(v) = 2. The signed graphs newlineadmitting an RDF are explored and certain classes of signed graphs such as paths, newlinecycles, stars admitting an RDF are characterised. Further, structural properties of signed graphs with 3-regular underlying graphs that admit an RDF are presented newlineand a characterisation of net-regular signed graphs with 3-regular underlying graphs, newlineadmitting an RDF is obtained. The signed graphs with Roman domination number equal to 2, 3, 4 and n are characterised. Further, criticality concepts have been examined by studying and#947;R-edge critical signed graphs S for which and#947;R(S +e) lt and#947;R(S), where the signature of the edge e is 1. A characterisation of and#947;R-edge critical signed trees with a single negative edge is presented, apart from some general results on and#947;R-edge critical signed graphs.&#13;
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