Graph Theory

Graph theory is a branch of mathematics that studies the properties of graphs, which are mathematical structures used to model pairwise relations between objects.

Basic Concepts

A graph is made up of a set of vertices (also called nodes or points) and a set of edges (also called links or lines). Each edge connects two vertices.

  1. Vertex: A vertex is a fundamental unit of a graph, representing an object or entity.

  2. Edge: An edge is a connection between two vertices, representing a relationship between the objects.

Types of Graphs

There are different types of graphs based on their properties:

  1. Undirected graph: A graph in which the edges have no direction, meaning that the connection between two vertices is bidirectional.

  2. Directed graph (Digraph): A graph in which the edges have a direction, meaning that the connection between two vertices is unidirectional.

  3. Weighted graph: A graph in which each edge has an associated numerical value, called its weight. Weighted graphs are commonly used to represent networks with varying connection strengths or costs.

  4. Complete graph: A graph in which every pair of distinct vertices is connected by an edge.

  5. Bipartite graph: A graph in which the vertex set can be partitioned into two disjoint sets such that no two vertices within the same set are adjacent.

Graph Properties and Algorithms

Graph theory provides several properties and algorithms for analyzing graphs:

  1. Degree: The degree of a vertex is the number of edges incident to the vertex. In a directed graph, we distinguish between the in-degree and the out-degree of a vertex.

  2. Path: A path is a sequence of vertices in a graph such that each vertex is connected to the next by an edge.

  3. Cycle: A cycle is a path in which the first and the last vertices are the same, and no vertex appears more than once.

  4. Connected graph: A graph is connected if there is a path between every pair of vertices.

  5. Graph traversal algorithms: Breadth-first search (BFS) and depth-first search (DFS) are common graph traversal algorithms used to visit all vertices of a graph in a systematic way.

  6. Shortest path algorithms: Dijkstra's algorithm and the Floyd-Warshall algorithm are used to find the shortest path between vertices in a graph.

  7. Minimum spanning tree: A minimum spanning tree is a tree that connects all the vertices in a weighted graph such that the total weight of the tree is minimized. Kruskal's algorithm and Prim's algorithm are used to find the minimum spanning tree of a graph.

Graph theory has numerous applications in computer science, engineering, biology, and other fields.