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Partial line graph operator and half-arc-transitive group actions
Marušič, Dragan ; Nedela, Roman
The concept of the partial line graph operator PI (and its inverse operator AI) on graphs of valency four with balanced orientation is developed in order to study transitive permutation groups having ...a non-self-paired suborbit of length 2 via the corresponding orbital graphs. If ▫$G$▫ is such a group and ▫$X$▫ is the orbital graph associated with a suborbit of length 2 of ▫$G$▫, which is not self-paired, then ▫$X$▫ has valency 4 and admits a vertex- and edge- but not arc-transitive action of ▫$G$▫. There is a natural balanced orientation of the edge set of ▫$X$▫ induced and preserved by ▫$G$▫. An analysis of the properties of this oriented graph is performed, using operators PI and Ai resulting in some partial results on the point stabilizer of ▫$G$▫ (in the case when ▫$X$▫ is connected). Finally, a graphical realization of such actions with nonabelian vertex stabilizers is given, that is, an infinite family of tetravalent graphs admitting a vertex and edge but not arc-transitive action with vertex stabilizer ▫$D_8$▫, the dihedral group of order 8, is constructed.
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