Adjacency matrix meaning and definition in DSA

An adjacency matrix is a square matrix of N x N size where N is the number of nodes in the graph and it is used to represent the connections between the vertices of a graph.

Characteristics of the adjacency matrix are:

• The size of the matrix is determined by the number of vertices (or nodes) in a graph.
• The edges in the graph are represented as values in the matrix. In case of unweighted graphs, the values are 0 or 1. In case of weighted graphs, the values are weights of the edges if edges are present, else 0.
• If the graph has few edges, the matrix will be sparse.

How to build an Adjacency Matrix:

It is very easy and simple to construct an adjacency matrix for a graph there are certain steps given below that you need to follow:

• Create an n x n matrix where n is the number of vertices in the graph.
• Initialize all elements to 0.
• For each edge (u, v) in the graph, if the graph is undirected mark a[u][v] and a[v][u] as 1, and if the edge is directed from u to v, mark a[u][v] as the 1. (Cells are filled with edge weight if the graph is weighted)

Applications of the Adjacency Matrix:

• Graph algorithms: Many graph algorithms like Floyd-Warshall algorithm
• Image processing: Adjacency matrices are used in image processing to represent the adjacency relationship between pixels in an image.
• Finding the shortest path between two nodes: By performing matrix multiplication on the adjacency matrix, one can find the shortest path between any two nodes in a graph. 

Advantages of using Adjacency Matrix:

• An adjacency matrix is simple and easy to understand.
• Adding or removing edges from a graph is quick and easy.
• It allows constant time access to any edge in the graph.

Disadvantages of using Adjacency Matrix:

• It is inefficient in terms of space utilisation for sparse graphs because it takes up O(N²) space.
• Computing all neighbors of a vertex takes O(N) time.

What else can you read?

• Kruskal’s Algorithm (Simple Implementation for Adjacency Matrix)
• Convert Adjacency Matrix to Adjacency List representation of Graph
• Convert Adjacency List to Adjacency Matrix representation of a Graph
• Comparison between Adjacency List and Adjacency Matrix representation of Graph