Adjacency matrix

In graph theory and computer science, 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 special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected, the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.

The adjacency matrix should be distinguished from the incidence matrix for a graph, a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and degree matrix which contains information about the degree of each vertex.

Definition

For a simple graph with vertex set V, the adjacency matrix is a square |V| × |V| matrix A such that its element Aij is one when there is an edge from vertex i to vertex j, and zero when there is no edge.[1] The diagonal elements of the matrix are all zero, since edges from a vertex to itself (loops) are not allowed in simple graphs. It is also sometimes useful in algebraic graph theory to replace the nonzero elements with algebraic variables.[2]

The same concept can be extended to multigraphs and graphs with loops by storing the number of edges between each two vertices in the corresponding matrix element, and by allowing nonzero diagonal elements. Loops may be counted either once (as a single edge) or twice (as two vertex-edge incidences), as long as a consistent convention is followed. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention.

Of a bipartite graph

The adjacency matrix A of a bipartite graph whose two parts have r and s vertices can be written in the form

where B is an r × s matrix, and 0r,r and 0s,s represent the r × r and s × s zero matrices. In this case, the smaller matrix B uniquely represents the graph, and the remaining parts of A can be discarded as redundant. B is sometimes called the biadjacency matrix.

Formally, let G = (U, V, E) be a bipartite graph with parts U = {u1, …, ur} and V = {v1, …, vs}. The biadjacency matrix is the r × s 0–1 matrix B in which bi,j = 1 if and only if (ui, vj)E.

If G is a bipartite multigraph or weighted graph then the elements bi,j are taken to be the number of edges between the vertices or the weight of the edge (ui, vj), respectively.

Variations

An (a, b, c)-adjacency matrix A of a simple graph has Ai,j = a if (i, j) is an edge, b if it is not, and c on the diagonal. The Seidel adjacency matrix is a (−1, 1, 0)-adjacency matrix. This matrix is used in studying strongly regular graphs and two-graphs.[3]

The distance matrix has in position (i, j) the distance between vertices vi and vj. The distance is the length of a shortest path connecting the vertices. Unless lengths of edges are explicitly provided, the length of a path is the number of edges in it. The distance matrix resembles a high power of the adjacency matrix, but instead of telling only whether or not two vertices are connected (i.e., the connection matrix, which contains boolean values), it gives the exact distance between them.