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Adjacency Algebra

The adjacency algebra of a coherent configuration with relations R_0, R_1, ..., R_d is the algebra spanned by their adjacency matrices A_0, A_1, ..., A_d. If p_(ij)^k are the intersection numbers, then

A_iA_j=sum_(k=0)^dp_(ij)^kA_k,

so the vector space span is closed under matrix multiplication. It is also closed under transpose and the Hadamard matrix product (Higman 1975).

For a homogeneous coherent configuration, the adjacency algebra is the Bose-Mesner algebra.

See also

Adjacency Matrix, Association Scheme, Bose-Mesner Algebra, Coherent Configuration, Homogeneous Coherent Configuration

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References

Higman, D. G. "Coherent Configurations. I. Ordinary Representation Theory." Geom. Dedicata 4, 1-32, 1975. https://doi.org/10.1007/BF00147398.

Cite this as:

Weisstein, Eric W. "Adjacency Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AdjacencyAlgebra.html

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