Adjacency Matrix in Data Structure and Algorithm

The adjacency matrix is one of the simplest and most widely used ways to represent graphs in data structures and algorithms. A graph is essentially a collection of objects (called vertices or nodes) that are connected by lines (called edges). Graphs help us model relationships in real-world systems such as social networks, computer networks, and transportation systems. To efficiently analyze and manipulate graphs, we need a clear representation, and the adjacency matrix offers a straightforward approach.

What is an Adjacency Matrix?

An adjacency matrix is a table, specifically a two-dimensional array, that tells us whether two vertices in a graph are connected. If there are \( V \) vertices in the graph, the matrix will have \( V imes V \) entries. Each row and column represent a vertex in the graph.

- For an **undirected graph** (where edges have no direction), the adjacency matrix is symmetric.
- For a **directed graph** (where edges have direction), the adjacency matrix can be asymmetric.
- If there is an edge between vertex \( i \) and vertex \( j \), the entry at \( M[i][j] \) will be \( 1 \) (or the weight of the edge in weighted graphs). If no edge exists, the entry will be \( 0 \).

How Does It Work?

Let’s break it down with an example:
Suppose we have 4 vertices labeled \( v_1, v_2, v_3, \) and \( v_4 \) and the following edges:
- \( v_1…