Adjacency algebra

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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[edit]

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 (ij)-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[edit]

  1. ^ a b c d Algebraic graph theory, by Norman L. Biggs, 1993, ISBN 0521458978, p. 9