Introduction to Graphs
Overview
A graph is a powerful tool for analyzing complex patterns and relationships, and it helps in drawing conclusions from them. It has wide applicability across various domains
A Brief Introduction to Graph
A graph is a well-defined collection of vertices (nodes) and edges (links/connections). Nodes generally represent attributes, features, objects, or items, whereas edges represent weights or relationships. A graph can be a network of points connected by edges. Applications of graphs can be found in various fields, including mathematics, computer science, chemistry, biology, social sciences, transportation, agriculture, operational research, etc.
There are various types of graphs, such as directed graphs, undirected graphs, multigraphs, planar graphs, complete graphs, simple graphs, weighted graphs, null graphs, finite graphs, infinite graphs, cyclic graphs, acyclic graphs, and many more. The images above describe two types of graphs:
- Directed Graph: A directed graph consists of edges with directions.
- Undirected Graph: An undirected graph consists of edges with no directions, allowing movement in any direction.
Let’s understand some other types as well:
1) Planar Graphs are graphs that can be drawn on a plane without any edges intersecting.
2) Multigraphs allow parallel edges.
3) Pseudographs are graphs that may contain both parallel edges and self-loops.
4) Complete Graphs are graphs where each node is connected to every other node, with all nodes being distinct.
5) Weighted Graphs are graphs where edges are assigned weights or values.
6) Null Graphs contain no edges.
7) Cyclic Graphs contain at least one cycle.
One of the important concepts in graph theory is the degree of a graph. The degree refers to the number of edges associated with a vertex. A pendant vertex is one with a degree of 1, while an isolated vertex has a degree of 0.
Now, let’s understand three very important theorems of graph theory:
- The sum of the degrees of all vertices in a graph is equal to twice the number of edges.
- The number of vertices of odd degree is even.
- The maximum degree of any vertex in a simple graph with mmm vertices is m−1.