Graph-Theory-The-Foundations-of-Modern-Networks.pptx

Graph Theory-The Foundations of Modern Networks

1. Graph Theory: The Foundations of Modern Networks Graph theory is the study of mathematical structures used to model pairwise relationships between objects. It provides a powerful framework for understanding and analyzing complex systems, with applications ranging from social networks to transportation planning. by Sitikant Behera

2. Introduction to Graphs: Nodes and Edges Nodes (Vertices) The fundamental building blocks of a graph, representing individual entities or objects. Edges The connections between nodes, representing the relationships or interactions between the entities. Graph The complete structure consisting of nodes and edges, modeling a complex system or network.

3. Graph Terminology: Adjacency, Degree, Connectivity 1 Adjacency Nodes are considered adjacent if they are connected by an edge. 2 Degree The number of edges connected to a node, indicating its level of connectivity. 3 Connectivity The extent to which nodes in a graph are interconnected, crucial for understanding network behavior.

4. Types of Graphs: Directed, Undirected, Weighted Directed Graphs Edges have a defined direction, representing one-way relationships. Undirected Graphs Edges have no defined direction, representing symmetric relationships. Weighted Graphs Edges have associated numerical values, capturing the strength or cost of relationships.

5. Graph Representations: Adjacency Matrices and Lists Adjacency Matrix A 2D array representing the connections between nodes, efficient for dense graphs. Adjacency List A collection of lists, each containing the neighbors of a node, efficient for sparse graphs. Tradeoffs The choice of representation depends on the graph's density and the specific algorithms being used.

6. Fundamental Graph Algorithms: Traversal, Shortest Path, Minimum Spanning Tree 1 Traversal Algorithms like Depth-First Search (DFS) and Breadth-First Search (BFS) for exploring graph structures. 2 Shortest Path Algorithms like Dijkstra's and Bellman-Ford for finding the optimal paths between nodes. 3 Minimum Spanning Tree Algorithms like Kruskal's and Prim's for identifying the most efficient set of connections.

7. Graph Applications: Social Networks, Recommendation Systems, Transportation Planning Social Networks Modeling relationships between users, enabling analysis of information diffusion and community detection. Recommendation Systems Leveraging graph structures to suggest relevant products, content, or connections to users. Transportation Planning Optimizing routes, schedules, and infrastructure using graph-based models of transportation networks.

8. Emerging Trends in Graph Theory: Big Data, Quantum Computing, Biological Networks Big Data Handling and analyzing massive, complex graphs to uncover insights in fields like social media and cybersecurity. Quantum Computing Leveraging the unique properties of quantum mechanics to solve challenging graph-related problems more efficiently. Biological Networks Modeling and understanding the intricate webs of interactions in biological systems, from cells to ecosystems.