Speaker：Binzhou Xia（The University of Melbourne，Australia）

Time：2022-11-16, 15:00

Location：Tencent Meeting ID：755-059-3114（No Password）

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 talk, we will give a solution to Babai's problem by constructing an infinite family of 2-arc-transitive digraphs and of vertex-primitive digraphs, respectively, both of whose adjacency matrices are non-diagonalizable.

This talk is based on joint work with Yuxuan Li, Sanming Zhou and Wenying Zhu.