文摘

A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be arc-transitive if its automorphism group acts transitively on the set of its arcs. All cubic and tetravalent arc-transitive bicirculants are known, and this paper gives a complete classification of connected pentavalent arc-transitive bicirculants. In particular, it is shown that, with the exception of seven particular graphs, a connected pentavalent bicirculant is arc-transitive if and only if it is isomorphic to a Cayley graph \(\mathrm{Cay}(D_{2n},\{b,ba,ba^{r+1},ba^{r^2+r+1},ba^{r^3+r^2+r+1}\})\) on the dihedral group \(D_{2n}=\langle a,b\mid a^n=b^2=baba=1 \rangle \) , where \(r\in \mathbb {Z}_n^*\) such that \(r^4+r^3+r^2+r+1 \equiv 0 \pmod {n}\) .