Adjacency algebra
From HandWiki
In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A.[1] Some other similar mathematical objects are also called "adjacency algebra".
Properties
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 (i, j)-th element of Ad.[1]
Statement. The dimension of the adjacency algebra of a connected graph of diameter d is at least d + 1.[1]
Corollary. A connected graph of diameter d has at least d + 1 distinct eigenvalues.[1]
References
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