On Edge-Transitive Graphs of Square-Free Order

Abstract

We study the class of edge-transitive graphs of square-free order and valency at most $k$. It is shown that, except for a few special families of graphs, only finitely many members in this class are basic (namely, not a normal multicover of another member). Using this result, we determine the automorphism groups of locally primitive arc-transitive graphs with square-free order.

Pentavalent semisymmetric graphs of square‐free order

“…Let $$N\le X\le G$$ with $$X\unicode{x02215}N\cong \text{soc}(\bar{G})$$. Then $${\rm{\Gamma }}$$ is $$X$$‐semisymmetric, and $$X=N\times T$$ by [[15], Theorem 30], where $$T$$ is a simple subgroup of $$X$$. In particular, $$T\cong \text{soc}(\bar{G})$$.…”

“…. Then Γ is X -semisymmetric, and X N T = × by [ [15], Theorem 30], where T is a simple subgroup of X . In particular, T G soc( )…”

A family of edge-transitive Cayley graphs

“…Moreover, edge-transitive graphs of square-free order and valency at most 7 have been classified by [9,17,19,20]. Quite recently, Li et al [18] characterized the 'basic' edge-transitive graphs (namely, each nontrivial normal subgroup of the edge-transitive automorphism group has at most two orbits on the vertex set) of square-free order, with certain cases needing further research. It seems difficult to approach a general classification of edgetransitive graphs of square-free order.…”