Adjacency matrix

In mathematics and computer science, a finite graph is often represented by its adjacency matrix. The adjacency matrix of a directed or undirected graph (with n vertices, say) is the n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex. (An alternative way to store graphs in computers, especially graphs with few edges, is given by the incidence matrix.)

The modified adjacency matrix is generated by replacing all entries greater than 1 in the adjacency matrix by 1.

Examples

The adjacency matrix for the example graph

is

${\begin{pmatrix}0&1&0&0&1&0\\1&0&1&0&1&0\\0&1&0&1&0&0\\0&0&1&0&1&1\\1&1&0&1&0&0\\0&0&0&1&0&0\\\end{pmatrix}}$

Properties

The adjacency matrix of undirected graphs is always symmetric.

Suppose two directed or undirected graphs G₁ and G₂ with adjacency matrices A₁ and A₂ are given. G₁ and G₂ are isomorphic if and only if there exists a permutation matrix P such that

PA₁P ⁻¹ = A₂.

In particular, A₁ and A₂ are similar and have therefore the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore serve as isomorphism invariants of graphs.

If A is the adjacency matrix of the directed or undirected graph G, then the matrix Aⁿ (i.e. the matrix product of n copies of A) has an interesting interpretation: the entry in row i and column j gives the number of (directed or undirected) paths of length n from vertex i to vertex j.

The matrix I−A (where I denotes the n-by-n identity matrix) is invertible if and only if there are no directed cycles in the graph G. In this case, the inverse (I−A)⁻¹ has the following interpretation: the entry in row i and column j gives the number of directed paths from vertex i to vertex j (which is always finite if there are no directed cycles). This can be understood using the geometric series for matrices:

(I−A)⁻¹ = I + A + A² + A³ + ...

corresponding to the fact that the number of paths from i to j equals the number of paths of length 0 plus the number of paths of length 1 plus the number of paths of length 2 etc.