A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p).
In this paper, a classification is given for tetravalent graphs of square-free order which are vertex-transitive and edge-transitive. It is shown that such graphs are Cayley graphs, edge-regular metacirculants and covers of some graphs arisen from simple groups A 7 , J 1 and PSL(2, p).
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.
Let G be a finite group with soc(G) = A c for c ≥ 5. A characterization of the subgroups with square-free index in G is given. Also, it is shown that a (G, 2)-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of 11 other graphs.2010 Mathematics subject classification: primary 20B15; secondary 20B30.
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