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''.
[Algebraic graph theory, by ]Norman L. Biggs
Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics..
Education
Biggs was educated at Harrow County Grammar School and then studied mathemat ...
, 1993,
p. 9
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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.][
]
References
Algebraic graph theory
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