Representation of Graph

A Graph is a non-linear data structure consisting of vertices and edges. The vertices are sometimes also referred to as nodes and the edges are lines or arcs that connect any two nodes in the graph. More formally a Graph is composed of a set of vertices(V) and a set of edges(E). The graph is denoted by G(V, E).

Representations of Graph

Here are the two most common ways to represent a graph : For simplicity, we are going to consider only unweighted graphs in this post.

1. Adjacency Matrix
2. Adjacency List

Adjacency Matrix Representation

An adjacency matrix is a way of representing a graph as a boolean matrix of (0's and 1's).

Let's assume there are n vertices in the graph So, create a 2D matrix adjMat[n][n] having dimension n x n.

• If there is an edge from vertex i to j, mark adjMat[i][j] as 1.
• If there is no edge from vertex i to j, mark adjMat[i][j] as 0.

Representation of Undirected Graph as Adjacency Matrix:

• We use an adjacency matrix to represent connections between vertices.
• Initially, the entire matrix is filled with 0s, meaning no edges exist.
• There is an edge between vertex 0 and vertex 1,so we set mat[0][1] = 1 and mat[1][0] = 1.
• There is an edge between vertex 0 and vertex 2,so we set mat[0][2] = 1 and mat[2][0] = 1.
• There is an edge between vertex 1 and vertex 2,so we set mat[1][2] = 1 and mat[2][1] = 1.

Adjacency Matrix Representation:
0 1 1 
1 0 1 
1 1 0 

Representation of Directed Graph as Adjacency Matrix:

• Initially, the entire matrix is filled with 0s, meaning no edges exist.
• Unlike an undirected graph, we do not set mat[destination][source] because the edge goes in only one direction.
• There is an edge between vertex 1 and vertex 0,so we set mat[1][0] = 1.
• There is an edge between vertex 2 and vertex 0,so we set mat[2][0] = 1.
• There is an edge between vertex 1 and vertex 2,so we set mat[1][2] = 1.

Adjacency Matrix Representation:
0 0 0 
1 0 1 
1 0 0 

Adjacency List Representation

An array of Lists is used to store edges between two vertices. The size of array is equal to the number of vertices (i.e, n). Each index in this array represents a specific vertex in the graph. The entry at the index i of the array contains a linked list containing the vertices that are adjacent to vertex i. Let's assume there are n vertices in the graph So, create an array of list of size n as adjList[n].

• adjList[0] will have all the nodes which are connected (neighbour) to vertex 0.
• adjList[1] will have all the nodes which are connected (neighbour) to vertex 1 and so on.

Representation of Undirected Graph as Adjacency list:

• We use an array of lists (or vector of lists) to represent the graph.
• The size of the array is equal to the number of vertices (here, 3).
• Each index in the array represents a vertex.
• Vertex 0 has two neighbours (1 and 2).
• Vertex 1 has two neighbours (0 and 2).
• Vertex 2 has two neighbours (0 and 1).

Adjacency List Representation:
0: 1 2 
1: 0 2 
2: 0 1 

Representation of Directed Graph as Adjacency list:

• We use an array of lists (or vector of lists) to represent the graph.
• The size of the array is equal to the number of vertices (here, 3).
• Each index in the array represents a vertex.
• Vertex 0 has no neighbours
• Vertex 1 has two neighbours (0 and 2).
• Vertex 2 has 1 neighbours (0).

Adjacency List Representation:
0: 
1: 0 2 
2: 0