Transitivity of Permuation Groups on Unordered Sets.
A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, all connected valency seven symmetric graphs of order are classified, where , are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order , and that for odd primes and , there is an infinite family of connected valency seven one-regular graphs of order with solvable automorphism groups, and there are four sporadic ones...
A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D n, E 6, E 7, E 8. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4.