PENTAVALENT SYMMETRIC GRAPHS OF ORDER $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit ...

A complete classification is given of pentavalent symmetric graphs of order $30p$, where $p\ge 5$ is a prime. It is proved that such a graph ${\Gamma }$ exists if and only if $p=13$ and, up to isomorphism, there is only one such graph. Furthermore, ${\Gamma }$ is isomorphic to $\mathcal{C}_{390}$, a coset graph of PSL(2, 25) with ${\sf Aut}\, {\Gamma }=\mbox{PSL(2, 25)}$, and ${\Gamma }$ is 2-regular. The classification involves a new 2-regular pentavalent graph construction with square-free order. [ABSTRACT FROM PUBLISHER]