Institute of Mathematics

of the Czech Academy of Sciences
 
 
 
 

Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 581-599, 2018

Valency seven symmetric graphs of order $2pq$

Xiao-Hui Hua, Li Chen

Received October 6, 2015.   Published online June 5, 2018.
Abstract:  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.
Keywords:  arc-transitive graph; symmetric graph; $s$-regular graph
Classification MSC:  05C25, 20B25
DOI:  10.21136/CMJ.2018.0530-15
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Affiliations:   Xiao-Hui Hua, Li Chen, College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, P. R. China, e-mail: xhhua@htu.cn
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