Adjacency algebra
Introduction 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''.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''.''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.
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