Bose–Mesner algebra
E886600
The Bose–Mesner algebra is a commutative matrix algebra arising from association schemes in algebraic combinatorics, fundamental for studying symmetric relations and distance-regular graphs.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
commutative algebra ⓘ finite-dimensional algebra ⓘ matrix algebra ⓘ |
| appearsIn | theory of symmetric relations on a finite set ⓘ |
| arisesFrom | association scheme ⓘ |
| closedUnder |
matrix addition
ⓘ
matrix multiplication ⓘ scalar multiplication ⓘ |
| contains |
all adjacency matrices of the association scheme
ⓘ
identity matrix ⓘ |
| context | finite association scheme ⓘ |
| dimensionEquals | number of associate classes in the association scheme ⓘ |
| encodes |
Krein parameters of an association scheme
ⓘ
eigenvalues of adjacency matrices ⓘ intersection numbers of an association scheme ⓘ |
| field |
complex numbers
ⓘ
real numbers ⓘ |
| generalizes | adjacency algebra of a regular graph ⓘ |
| hasApplication |
analysis of linear codes
ⓘ
classification of distance-regular graphs ⓘ construction of combinatorial designs ⓘ eigenvalue bounds for graphs ⓘ |
| hasBasis |
adjacency matrices of an association scheme
ⓘ
primitive idempotents of an association scheme ⓘ |
| hasDecomposition | simultaneous eigenspace decomposition ⓘ |
| hasDualBasis | primitive idempotent basis ⓘ |
| hasDualStructureConstants | Krein parameters ⓘ |
| hasProperty |
all basis adjacency matrices are 0–1 matrices
ⓘ
basis adjacency matrices are pairwise disjoint in support ⓘ basis adjacency matrices sum to the all-ones matrix ⓘ |
| hasStructureConstants | intersection numbers ⓘ |
| isCommutative | true ⓘ |
| isSemisimple | true ⓘ |
| isSimultaneouslyDiagonalizable | true ⓘ |
| namedAfter |
D. M. Mesner
NERFINISHED
ⓘ
R. C. Bose NERFINISHED ⓘ |
| relatedTo |
Terwilliger algebra
NERFINISHED
ⓘ
distance-regular graph ⓘ strongly regular graph ⓘ symmetric association scheme ⓘ |
| typicalReference |
Bannai–Ito theory of association schemes
NERFINISHED
ⓘ
Brouwer–Cohen–Neumaier distance-regular graphs NERFINISHED ⓘ |
| usedIn |
algebraic combinatorics
ⓘ
coding theory ⓘ design theory ⓘ spectral graph theory ⓘ study of distance-regular graphs ⓘ theory of association schemes ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.