Graph theory all basic concepts ... - tlooto, The Most Powerful AcademicGPT

Graph theory is a fundamental branch of mathematics concerned with the study of graphs, which mathematically model pairwise relations among a set of objects. It provides a versatile toolbox extensively applied in various fields including computer science, biology, ecology, neuroscience, and social sciences, enabling the analysis of complex networks ranging from brain connectivity to landscape ecology [1][2][3]. --- 1. Basic Definition of a Graph A graph $ G $ is defined as an ordered pair: $$ G = (V, E) $$ where: - $ V $ is the finite set of vertices (also called nodes or points), - $ E \subseteq \{ \{u,v\} | u,v \in V, u \neq v \} $ is the set of edges (unordered pairs of vertices representing connections). In the case of directed graphs, edges are ordered pairs $ (u,v) $, representing a connection from vertex $ u $ to $ v $. --- 2. Types of Graphs - Undirected Graph: Edges have no direction; edge $ (u,v) $ is identical to $ (v,u) $. - Directed Graph (Digraph): Edges h