  The fact that the adjacency matrix of every finite graph is diagonalizable
plays a fundamental role in spectral graph theory. Since this fact does not
hold in general for digraphs, it is natural to ask whether it holds for
digraphs with certain level of symmetry. Interest on this question dates back
to early 1980s, when P.~J.~Cameron asked for the existence of arc-transitive
digraphs with non-diagonalizable adjacency matrix. This was answered in the
affirmative by L.~Babai in 1985. Then Babai posed the open problem of
constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose
adjacency matrices are not diagonalizable. In this paper, we solve Babai's
problem by constructing an infinite family of $s$-arc-transitive digraphs for
each integer $s\geq2$, and an infinite family of vertex-primitive digraphs,
respectively, both of whose adjacency matrices are non-diagonalizable.
