Adjacency algebra
In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A.
Some other similar mathematical objects are also called "adjacency algebra".Properties of the adjacency algebra of G are associated with various spectral, adjacency and connectivity properties of G.
Statement. The number of walks of length d between vertices i and j is equal to the -th element of Ad.
Statement. The dimension of the adjacency algebra of a connected graph of diameter d is at least d + 1.
Corollary. A connected graph of diameter d has at least d + 1 distinct eigenvalues.
