Classification of two-distance-transitive Cayley graphs of the semi-dihedral groups

The class of 2-distance-transitive graphs naturally generalizes distance-transitive graphs and plays a central role in algebraic graph theory. Classifying such graphs for a prescribed underlying group is a key open problem. A vertex-transitive graph $\mathrm{\Gamma }$ is said to be $2$-distance-transitive if, for each $i\in \{1,2\}$, any two pairs of vertices with identical distance $i$ in $\mathrm{\Gamma }$ can be mapped to each other via some automorphism of the graph. In this paper, we present a complete classification of all $2$-distance-transitive Cayley graphs of the semi-dihedral groups.