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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 nonprincipal 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. [ABSTRACT FROM AUTHOR]

A k‐regular graph is called a divisible design graph (DDG) if its vertex set can be partitioned into m classes of size n, such that two distinct vertices from the same class have exactly λ1 common neighbours, and two vertices from different classes have exactly λ2 common neighbours. A DDG with m=1, n=1, or λ1=λ2 is called improper, otherwise it is called proper. We present new constructions of DDGs and, using a computer enumeration algorithm, we find all proper connected DDGs with at most 39 vertices, except for three tuples of parameters: (32,15,6,7,4,8), (32,17,8,9,4,8), and (36,24,15,16,4,9). [ABSTRACT FROM AUTHOR]