A means of representing which vertices (or nodes) of a graph are adjacent to which other vertices.
The adjacency matrix of a finite graph G on n vertices is the n×n matrix where the non-diagonal entry aij is the number of edges from vertex i to vertex j, and the diagonal entry aii, depending on the convention, is either once or twice the number of edges (loops) from vertex i to itself.
Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention.
There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs.
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.
