Graph Theory

Graph theory is a mathematical concept that deals with the study of Graphs, which are structures consisting of vertices (nodes) connected by edges (links). These graphs are used to model relationships and connections between various entities.
In graph theory, the analysis and manipulation of graphs are used to solve a wide range of real-world problems in diverse fields such as computer science, mathematics, social sciences, and engineering and numerous practical applications such including network analysis, social network modeling, routing algorithms, scheduling problems, and optimization in various domains.

Components

Graphs consist of Vertices, which represent the entities, and Edges, which represent the connections or relationships between the entities.
Paths refer to seqrtices connected by edges, while cycaths that form a closed loop, starting and ending at the same vertex.


Types of Graphs:

  • Directed Graph (Digraph): A graph in which the edges have a specific direction, indicating a one-way relationship between vertices.
  • Undirected Graph: A graph in which the edges have no specific direction, representing a two-way relationship between vertices.
  • Weighted Graph: A graph in which each edge has an associated weight or cost, often used to represent distances, capacities, or other quantitative measures.
  • Complete Graph: A graph in which each pair of distinct vertices is connected by a unique edge, resulting in a fully connected graph.
  • Bipartite Graph: A graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent.
  • Cyclic Graph: A graph that contains at least one cycle, forming a closed loop of edges and vertices.
  • Acyclic Graph: A graph that does not contain any cycles, also known as a directed acyclic graph (DAG) in the case of a directed graph.
  • Planar Graph: A graph that can be embedded in the plane without any edge intersections, often used in geographical and map-related applications.
  • Complete Bipartite Graph: A bipartite graph in which every vertex in one set is connected to every vertex in the other set.
  • Regular Graph: A graph in which each vertex has the same degree, representing the number of edges incident to the vertex.