A Review of Graph Theory and Its Applications Across Various Disciplines by IRJET Journal

International Research Journal of Engineering and Technology (IRJET)

e-ISSN: 2395-0056

Volume: 11 Issue: 03 | Mar 2024

p-ISSN: 2395-0072

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A Review of Graph Theory and Its Applications Across Various Disciplines Qurratulain Mevegar1, Dr. Asha Saraswathi B2 , Waseem Ahmed Halwegar3 1 Assistant Professor, Department of Mathematics, Anjuman Institute of Technology and Management, Bhatkal,

Karnataka, India.

2 Associate Professor & HOD, Department of Mathematics, Institute of Engineering and Technology, Srinivas

University. Mangalore, India

3Assitant Professor & HOD, Department of Basic Sciences, Anjuman Institute of Technology and Management

(AITM), Bhatkal, Karnataka, India. --------------------------------------------------------------------------***----------------------------------------------------------------------ABSTRACT This review paper offers a comprehensive analysis how graph theory may be applied to solve problems in a variety of fields, including computer science, environmental geography, power systems, and medical applications. The main goals of power systems evaluation are reliability improvement and vulnerability assessment through the use of game theory and graph theory models. In computer science, novel approaches to network optimisation and cybersecurity reinforcement are demonstrated by IoT-based machine learning, graph neural networks, and link prediction. Studies in medicine demonstrate how important graph theory is to understanding the relationships between brain shape and function, identifying cancer, and understanding how brain networks related to pain are organised. Graph theory is important for landscape connectivity, genetic diversity, and ecological network design in environmental applications because it sheds light on ecological structures, the effects of watershed development, and spatial patterns. Overall, the review highlights graph theory's wide significance and prospective research directions across disciplinary boundaries.

1. INTRODUCTION: Graph theory is a significant field in Discrete Mathematics and serves as an important area of mathematical coordination. The modelling of objects as vertices and the representation of relationships as edges, offering a powerful approach to modelling complex situations (Majeed & Rauf, 2020). Graphs are critical for explaining complicated structures, from sports competitions like football tournaments to the complexities of social networks and the Internet as a whole (Miz et al., 2019). Graph theory is a versatile technique that can be used to map social relationships between individuals or to define sports teams and their results (Duque, ML Martins & Manuel Clemente, 2016). In essence, it provides a mathematical foundation and serves as a strong analytical framework for understanding and modelling complex, interconnected systems across multiple domains (Duindam et al., 2009). In the field of computer science research, graph theory has numerous applications in areas such as data mining, image capture, segmentation, networking, and clustering (Ahmed, 2012). It is useful for practical tasks including mail delivery route optimisation, troubleshooting and correcting network issues, and topologically based strategic local area network (LAN) planning (Makeri, 2019). Complex problems such as understanding the dynamics of bilateral institutions and negotiating the complexities of the connections between employers and job seekers are addressed by scientific research into graph theory (Ramalingam et al., 2008). Graph theory plays a key role in directing marketing tactics and advancing both business and agriculture (King et al., 2010). Following the principles of graph theory, individual points are called vertices, and the edges—the connections between them—specify the relationships between them (Wilson, 1979). This methodical abstraction enables a comprehensive analysis of interrelated components, offering a rigorous and useful analytical tool for strategic decisionmaking in the fields of business and agricultural growth (Mollinga & Gondhalekar, 2014). “Fig. 1.” is a simple graph. Here we can see in the graph G consists of a no empty finite set. 𝑉(𝐺) of segments called vertices (nodes) and a finite set 𝐸(𝐺)of distinct unordered pairs of distinct elements of 𝑉(𝐺) called edge. A graph is a join 𝐺 = (𝑉, 𝐸)of sets of fulfilment 𝐸 ⊆ ,𝑣-2; We will always take 𝑉 ∩ 𝐸 = ∅ to avoid the ambiguity of notation. The Elements

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