Abstract: A graph is said to be stable if its canonical double cover has no unexpected symmetries. Graph

stability has been studied in the literature from different viewpoints. In this talk I will first review these

viewpoints and then focus on the stability of circulant graphs. In particular, I will give an answer to a

question of Wilson in 2008 on the stability of arc-transitive circulant graphs and infinitely many

counterexamples to a conjecture of Marusic, Scapellato and Zagaglia Salvi in 1989. This is based

on joint work with Yan-Li Qin and Sanming Zhou.