The undirected Bruhat graphΓ(u,v)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (u,v)$$\end{document} has the elements of the Bruhat interval [u, v] as vertices, with edges given by multiplication by a reflection. Famously, Γ(e,v)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (e,v)$$\end{document} is regular if and only if the Schubert variety Xv\documentclass[12pt]{minimal}
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\begin{document}$$X_v$$\end{document} is smooth, and this condition on v is characterized by pattern avoidance. In this work, we classify when Γ(e,v)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (e,v)$$\end{document} is vertex-transitive; surprisingly this class of permutations is also characterized by pattern avoidance and sits nicely between the classes of smooth permutations and self-dual permutations. This leads us to a general investigation of automorphisms of Γ(u,v)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (u,v)$$\end{document} in the course of which we show that special matchings, which originally appeared in the theory of Kazhdan–Lusztig polynomials, can be characterized, for the symmetric and right-angled groups, as certain Γ(u,v)\documentclass[12pt]{minimal}
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\begin{document}$$\Gamma (u,v)$$\end{document}-automorphisms which are conjecturally sufficient to generate the orbit of e under Aut(Γ(e,v))\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{Aut}\,}}(\Gamma (e,v))$$\end{document}.
