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An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices in the graph are adjacent or not. For a graph with n vertices, the adjacency matrix is an n x n matrix A, where A is defined as:

A[i][j] = 1 if there is an edge between vertex i and vertex j, and 0 otherwise.

In simple undirected graphs, the adjacency matrix is symmetric, meaning A[i][j] = A[j][i]. If the graph has loops (edges connecting a vertex to itself), the diagonal elements A[i][i] can be non-zero. For directed graphs, the matrix is not necessarily symmetric, as an edge from i to j does not imply an edge from j to i.

The number of edges in an undirected graph is half the sum of all elements in its adjacency matrix. For a directed graph, the sum of all elements is the total number of edges. The adjacency matrix is a fundamental tool in graph theory and computer science for algorithms such as graph traversal, connectivity analysis, and shortest path calculations. It provides a direct way to check for the existence of an edge between any two vertices in constant time, O(1). However, for sparse graphs (graphs with relatively few edges), adjacency lists are often preferred due to lower memory requirements.