Minimum supports of eigenfunctions in bilinear forms graphs

In this paper we study eigenfunctions corresponding to the minimum eigenvalue of bilinear forms graphs. Our main goal is to find eigenfunctions with the supports (non-zero positions) of minimum cardinality. For bilinear forms graphs of diameter $D=2$ over a prime field we prove that there exist eigenfunctions with the support achieving the weight distribution bound. We also provide an explicit construction of such functions. For bilinear forms graphs of diameter $D\ge 3$ we show the non-existance of eigenfunctions with supports achieving the weight distribution bound.