Pentavalent vertex-transitive diameter two graphs
This paper considers the class of \(5\)-valent, vertex-transitive graphs of diameter \(2\). For a graph \(\Gamma\) in this class with automorphism group \(A\), we call \(\Gamma\) distance-transitive if \(u\in V\Gamma\) and the stabilizer of \(u\) in \(A\) acts transitively both on the set of vertices adjacent to \(u\) and on the set of vertices at distance \(2\) from \(u\). It is shown that the graphs in this class that are not distance-transitive are exactly the following: \(\overline{C_8},\;\overline{2C_4}\), the Cartesian products \(K_5\square K_2\) and \(K_3\square K_4\), and the lexicographic product \(C_5[K_2]\). The distance-transitive graphs of valence \(5\) had been previously determined by \textit{A. E. Brouwer} et al. [Distance-regular graphs. Berlin etc.: Springer-Verlag (1989; Zbl 0747.05073)].
scientific article; zbMATH DE number 6823735
scientific article