Types of Graphs with Examples

A graph is a mathematical structure used to represent relationships between objects. It consists of:

• Vertices (or nodes): The points in the graph representing entities.
• Edges: The lines connecting pair of vertices, representing relationships or interactions.

Graphs can be classified in multiple ways based on their properties. Here’s a structured categorization:

Based On Size:

Finite Graphs

A finite graph is a graph with a finite number of vertices and edges. In other words, both the number of vertices and the number of edges in a finite graph are limited and can be counted. Finite graphs are used to represent real-world situations where there is a limited number of objects and their connections. They help in organizing, analyzing, and optimizing relationships in different applications.

Infinite Graph: 

A graph is called an infinite graph if it has an infinite number of vertices and an infinite number of edges. Unlike finite graphs, which have a fixed number of nodes and connections, infinite graphs extend indefinitely.

Based on Structure:

Trivial Graph

A graph is said to be trivial if a finite graph contains only one vertex and no edge. It is also known as a singleton graph or a single vertex graph. A trivial graph is the simplest type of graph and is often used as a starting point for building more complex graphs. In graph theory, trivial graphs are considered to be a degenerate case and are not typically studied in detail

Simple Graph

A simple graph is a graph that does not contain more than one edge between the pair of vertices. A simple railway track connecting different cities is an example of a simple graph. 

Multi Graph

Any graph which contains some parallel edges but doesn’t contain any self-loop is called a multigraph. For example a Road Map. 

• Parallel Edges: If two vertices are connected with more than one edge then such edges are called parallel edges that are many routes but one destination.
• Loop: An edge of a graph that starts from a vertex and ends at the same vertex is called a loop or a self-loop.

Null Graph

A graph of order n and size zero is a graph where there are only isolated vertices with no edges connecting any pair of vertices.A null graph is a graph with no edges. In other words, it is a graph with only vertices and no connections between them. A null graph can also be referred to as an edgeless graph, an isolated graph, or a discrete graph

Complete Graph

A simple graph with n vertices is called a complete graph if the degree of each vertex is n-1, that is, one vertex is attached with n-1 edges or the rest of the vertices in the graph. A complete graph is also called Full Graph. 

Based On Direction:

Directed Graphs:

A graph in which edges have a direction, i.e., the edges have arrows indicating the direction of traversal.

Undirected Graphs

An undirected graph is a graph where edges do not have a specific direction, meaning connections go both ways. If two places are connected, you can travel in either direction. Examples include friendships on social media and two-way roads.

Based on Edge Weights:

Weighted Graphs

A weighted graph is a graph where each edge has a number (weight) that represents distance, cost, or time. These graphs help find the shortest or cheapest paths. Examples include Google Maps, airline routes, and delivery networks.

Unweighted Graphs

An unweighted graph is a graph where all edges are treated equally, with no extra values like distance or cost. It simply shows connections between points. Examples include basic social networks and metro maps without travel times.

Special Graph:

Pseudo Graph

A pseudograph is a type of graph that allows for the existence of self-loops (edges that connect a vertex to itself) and multiple edges (more than one edge connecting two vertices). In contrast, a simple graph is a graph that does not allow for loops or multiple edges. 

Regular Graph

A regular graph is a type of undirected graph in which every vertex has the same number of edges (or neighbors). In other words, all vertices in a regular graph have the same degree.

Based On Density:

Sparse Graphs

A graph with relatively few edges compared to the number of vertices. Example: A chemical reaction graph where each vertex represents a chemical compound and each edge represents a reaction between two compounds.

Dense Graphs

A graph with many edges compared to the number of vertices. Example: A social network graph where each vertex represents a person and each edge represents a friendship.

Based on Connectivity:

Connected or Disconnected Graph

Graph is said to be connected if there exists at least one path between each and every pair of vertices in graph G, otherwise, it is disconnected. A null graph with n vertices is a disconnected graph consisting of n components. Each component consists of one vertex and no edge. 

Based On Cycles:

Cyclic Graph

A graph G consisting of n vertices and n> = 3 that is V1, V2, V3- - - - Vn and edges (V1, V2), (V2, V3), (V3, V4)- - - - (Vn, V1) are called cyclic graph. 

Trees

A tree is a connected graph that contains no cycles. In other words, there is exactly one path between any two vertices.