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Adjacency matrix meaning and definition in DSA

Last Updated : 01 Aug, 2023
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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 edges of a graph.

Graph representation of undirected graph to Adjacency Matrix

Characteristics of the adjacency matrix are:

  • The size of the matrix is determined by the number of vertices in the graph.
  • The number of nodes in the graph determines the size of the matrix.
  • The number of edges in the graph is simply calculated.
  • 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:

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.

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