Pentavalent symmetric graphs of order 30p

Abstract

A complete classification is given of pentavalent symmetric graphs of order 30 p, where p ≥ 5 is a prime. It is proved that such a graph Γ exists if and only if p = 13 and, up to isomorphism, there is only one such graph. Furthermore, Γ is isomorphic to C390, a coset graph of PSL(2, 25) with AutΓ = PSL(2, 25), and Γ is 2-regular. The classification involves a new 2-regular pentavalent graph construction with square-free order.