Adjacency matrix

Adjacency Matrix

An adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the context of an undirected graph, the adjacency matrix is symmetric.

Definition

For a simple graph with n vertices, the adjacency matrix is an n _ n matrix A where the non-diagonal entry a_ij is the number of edges from vertex i to vertex j, and the diagonal entry a_ii is the number of loops at vertex i.

In the case of an undirected graph, the adjacency matrix is symmetric. If the graph is weighted, the elements of the matrix are the weights of the edges.

Properties

• The adjacency matrix of a simple graph is a binary matrix.
• The degree of a vertex in a graph can be determined by summing the entries in the corresponding row (or column) of the adjacency matrix.
• The adjacency matrix of a complete graph has all off-diagonal entries equal to 1.
• The adjacency matrix of a bipartite graph can be permuted to have a block form with zero diagonal blocks.

Applications

Adjacency matrices are used in various applications, including:

• Graph theory
• Network analysis
• Computer science algorithms
• Social network analysis

Examples

Symmetric Group 4

The symmetric group on 4 elements, denoted as S₄, can be represented by various Cayley graphs. Below are examples of adjacency matrices for different Cayley graphs of S₄:

• Cayley graph with generators {1, 5, 21}:

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Adjacency matrix for Cayley graph with generators {1, 5, 21}

• Cayley graph with generators {4, 9}:

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Adjacency matrix for Cayley graph with generators {4, 9}

Related Pages

• Graph (discrete mathematics)
• Incidence matrix
• Laplacian matrix
• Adjacency list

Gallery

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  Example of a graph represented by an adjacency matrix

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  Cayley graph with generators {1, 5, 21}

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  Cayley graph with generators {4, 9}

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  Adjacency matrix of a 6-node graph

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  Cayley graph of the symmetric group S4 with generators 1, 5, 21 (Nauru Petersen)

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  Adjacency matrix of the Cayley graph of S4 with generators 1, 5, 21

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  Cayley graph of the symmetric group S4 with generators 4, 9

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  Adjacency matrix of the Cayley graph of S4 with generators 4, 9