A Solution to Babai’s Problems on Digraphs with Non-diagonalizable Adjacency Matrix

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 in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems 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 problems by constructing an infinite family of s-arc-transitive digraphs for each integer \(s\ge 2\), and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable.