  This paper lies in the context of the studies of eigenfunctions of graphs
having minimum cardinality of support. One of the tools is the
weight-distribution bound, a lower bound on the cardinality of support of an
eigenfunction of a distance-regular graph corresponding to a non-principal
eigenvalue. The tightness of the weight-distribution bound was previously shown
in general for the smallest eigenvalue of a Grassmann graph. However, a
characterisation of optimal eigenfunctions was not obtained. Motivated by this
open problem, we consider the class of strongly regular Grassmann graphs and
give the required characterisation in this case. We then show the tightness of
the weight-distribution bound for block graphs of affine designs (defined on
the lines of an affine space with two lines being adjacent when intersect) and
obtain a similar characterisation of optimal eigenfunctions.
