Graph Theory - Adjacency Matrix

Quiz

Adjacency Matrix

An adjacency matrix is a square matrix used to represent a graph. It is useful for representing graphs where it is important to know whether two vertices are adjacent (i.e., there is an edge between them). The adjacency matrix provides an efficient way to store graph information and check for edges between vertices.

For an undirected graph, the adjacency matrix is symmetric, as an edge from vertex i to vertex j implies an edge from vertex j to vertex i.

Adjacency Matrix:
[[0 0 1 1 0 0]
 [0 0 0 0 1 1]
 [1 0 0 0 1 0]
 [1 0 0 0 0 0]
 [0 1 1 0 0 0]
 [0 1 0 0 0 0]]

Structure of the Adjacency Matrix

An adjacency matrix is a 2D array or matrix where each element matrix[i][j] represents the presence (or absence) of an edge between the vertices i and j. The dimensions of the matrix are n x n, where n is the number of vertices in the graph.

If there is an edge between vertex i and vertex j, the value at matrix[i][j] is 1 (or the weight of the edge if the graph is weighted). If no edge exists, the value is 0. For weighted graphs, the matrix stores the weight of the edge instead of 1.

Adjacency Matrix for Undirected Graphs

For an undirected graph, the adjacency matrix is symmetric, meaning that if vertex i is connected to vertex j, then matrix[i][j] = matrix[j][i].

Consider an undirected graph with 3 vertices: V = {A, B, C}, and edges {A-B, A-C}. The adjacency matrix for this graph would look like this −

Adjacency Matrix for Directed Graphs

For a directed graph, the adjacency matrix is not necessarily symmetric. If there is an edge from vertex i to vertex j, the matrix stores a value of 1 at matrix[i][j] but not necessarily at matrix[j][i].

Consider a directed graph with 3 vertices: V = {A, B, C}, and edges {A->B, A->C}. The adjacency matrix for this directed graph would look like this −

Adjacency Matrix for Weighted Graphs

In a weighted graph, the adjacency matrix stores the weight of the edge between vertices instead of just 1 or 0. If the weight between vertices i and j is w, then matrix[i][j] = w.

Consider a weighted graph with 3 vertices: V = {A, B, C}, and edges {A-B: 5, A-C: 3}. The adjacency matrix for this weighted graph would look like this −

Properties of the Adjacency Matrix

The adjacency matrix has several major properties that make it useful for graph representation, they are −

Space Complexity

The space complexity of an adjacency matrix is O(n), where n is the number of vertices in the graph. This is because the matrix stores information for every possible pair of vertices, regardless of whether an edge exists between them.

Edge Lookup

Checking whether there is an edge between two vertices can be done in constant time, O(1). You can simply access the matrix at matrix[i][j] to determine if an edge exists.

Memory Usage

The adjacency matrix can be inefficient in terms of memory usage, especially for sparse graphs (graphs with fewer edges). For sparse graphs, the matrix will still require storage for all n possible edges, which can waste memory.

Edge Insertion

Inserting an edge in an adjacency matrix can be done in constant time, O(1). You can directly set the appropriate element in the matrix to 1 (or the edge weight).

Traversal Complexity

Traversal of an adjacency matrix typically requires iterating over the rows or columns, which has a time complexity of O(n) for the entire graph. This can be slow for large graphs.

Handling Loops

In a graph, a loop is an edge that connects a vertex to itself. In an adjacency matrix, loops are represented by setting matrix[i][i] to 1 (or the weight of the loop). The presence of loops can be easily detected by checking the diagonal elements of the matrix.

Handling Multiple Edges (Multigraphs)

For multigraphs (graphs with multiple edges between the same pair of vertices), the adjacency matrix can store the number of edges or the total weight of all edges between two vertices. Instead of 1, the matrix stores the count or weight of the edges.

Applications of the Adjacency Matrix

The adjacency matrix is commonly used in various graph-related algorithms and applications, such as −

Graph Traversal

Graph traversal algorithms such as Depth-First Search (DFS) and Breadth-First Search (BFS) can be implemented efficiently with adjacency matrices. While the time complexity of these algorithms remains O(n) for an adjacency matrix, the simplicity of edge lookups makes it a convenient representation for smaller graphs.

Finding Shortest Paths

The adjacency matrix is used in algorithms like Floyd-Warshall to find the shortest paths between all pairs of vertices. The matrix stores the direct distances between vertices, and the algorithm iteratively updates it to find the shortest paths.

Graph Connectivity

The adjacency matrix can be used to determine if a graph is connected. If there exists a path from every vertex to every other vertex, the graph is connected. This can be checked using graph traversal techniques such as DFS or BFS.

Graph Algorithms

Various other graph algorithms, including those for detecting cycles, topological sorting, and checking bipartiteness, can also utilize adjacency matrices for efficient computation.

Matrix Operations for Graphs

In linear algebra, matrix operations such as matrix multiplication can be used to analyze graph properties. For example, the power of the adjacency matrix can be used to find paths of length 2, 3, or more between vertices.

Adjacency Matrix: Pros and Cons

The adjacency matrix has several advantages and disadvantages depending on the graph's structure and the algorithms being applied −

Advantages

Following are the advantages of adjacency matrix −

Simple to Implement: The adjacency matrix is easy to implement and access, making it a good choice for small to medium-sized graphs.
Efficient Edge Lookup: The matrix allows for constant-time edge lookups, making it useful for algorithms that need to quickly check the presence of an edge.
Good for Dense Graphs: The adjacency matrix is well-suited for dense graphs where the number of edges is close to the number of possible edges (O(n)).

Disadvantages

Following are the disadvantages of adjacency matrix −

Space Inefficient for Sparse Graphs: For sparse graphs, the adjacency matrix consumes a lot of memory to store information about edges that do not exist.
Slow for Large Graphs: The time complexity for traversal and manipulation of large graphs can be prohibitive when using an adjacency matrix, as it involves O(n) operations.
Redundant Information: The matrix stores information about every possible edge, even if many edges do not exist, leading to redundant memory usage.

Adjacency Matrix Alternatives

In practice, there are several alternatives to the adjacency matrix, especially for large and sparse graphs −

Adjacency List

The adjacency list is a more space-efficient representation for sparse graphs. It uses less memory and provides faster traversal for sparse graphs, as it only stores the edges that exist in the graph.

Edge List

An edge list is a simple representation where each edge is stored as a pair of vertices. It is space-efficient for very sparse graphs but does not support efficient edge lookups.

Compressed Adjacency Matrix

For large graphs with many missing edges, compressed representations of the adjacency matrix, such as sparse matrices or compressed sparse row (CSR) format, can be used to save space and improve efficiency.

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