There is a type of distance-regular graph, said to be $Q$-polynomial. In this paper we investigate a generalized $Q$-polynomial property involving a graph that is not necessarily distance-regular. We give a detailed description of an example associated with the projective geometry $L_N(q)$.
Projective geometries, $Q$-polynomial structures, and quantum groups
Distance-regular graphs and the $q$-tetrahedron algebra
Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property
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