A solution to Babai's problem on digraphs with non-diagonalizable adjacency matrix

TL;DR

This paper constructs infinite families of highly symmetric digraphs with non-diagonalizable adjacency matrices, addressing a long-standing open problem in spectral graph theory related to digraph symmetry and matrix diagonalizability.

Contribution

It provides the first explicit constructions of infinite families of s-arc-transitive and vertex-primitive digraphs with non-diagonalizable adjacency matrices.

Findings

Constructed infinite s-arc-transitive digraphs with non-diagonalizable matrices for all s≥2.

Constructed infinite vertex-primitive digraphs with non-diagonalizable matrices.

Resolved Babai's open problem from 1985.

Abstract

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.