On regular induced subgraphs of generalized polygons

For prime powers q, we also improve the known upper bounds on c(q,8) and c(q,12) by giving new geometric constructions of q-regular induced subgraphs in the symplectic generalized quadrangle W(3,q) and the split Cayley hexagon H(q), respectively. Our constructions show thatc(q,8)⩽2(q3−qq−q) for q an even power of a prime, andc(q,12)⩽2(q5−3q3) for all prime powers q. For q∈{3,4,5} we also give a computer classification of all q-regular induced subgraphs of the classical generalized quadrangles of order q. For W(3,7) we classify all 7-regular induced subgraphs which have a non-trivial automorphism.