Valency seven symmetric graphs of order \(2pq\). (English) Zbl 1488.05234

Hua, Xiao-Hui; Chen, Li

Czech. Math. J. 68, No. 3, 581-599 (2018).

Summary: 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 \(2pq\) are classified, where \(p\), \(q\) are distinct primes. It follows from the classification that there is a unique connected valency seven symmetric graph of order \(4p\), and that for odd primes \(p\) and \(q\), there is an infinite family of connected valency seven one-regular graphs of order \(2pq\) with solvable automorphism groups, and there are four sporadic ones with nonsolvable automorphism groups, which is \(1,2,3\)-arc transitive, respectively. In particular, one of the four sporadic ones is primitive, and the other two of the four sporadic ones are bi-primitive.

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures

Keywords:

arc-transitive graph; symmetric graph; \(s\)-regular graph