Every circular Hessenberg pair on a finite-dimensional vector space satisfies the two tridiagonal relations.
Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property
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This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the $Q$-polynomial property.
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Circular Hessenberg pairs and the tridiagonal relations
Every circular Hessenberg pair on a finite-dimensional vector space satisfies the two tridiagonal relations.