Edge-transitive bi-p-metacirculants of valency p
Keywords:Bi-p-metacirculant, edge-transitive, inner-abelian p-group
Let p be an odd prime. A graph is called a bi-p-metacirculant on a metacyclic p-group H if admits a metacyclic p-group H of automorphisms acting semiregularly on its vertices with two orbits. A bi-p-metacirculant on a group H is said to be abelian or non-abelian according to whether or not H is abelian.
By the results of Malnič et al. in 2004 and Feng et al. in 2006, we see that up to isomorphism, the Gray graph is the only cubic edge-transitive non-abelian bi-p-metacirculant on a group of order p3. This motivates us to consider the classification of cubic edge-transitive bi-p-metacirculants. Previously, we have proved that a cubic edge-transitive non-abelian bi-p-metacirculant exists if and only if p = 3. In this paper, we give a classification of connected edge-transitive non-abelian bi-p-metacirculants of valency p, and consequently, we complete the classification of connected cubic edge-transitive non-abelian bi-p-metacirculants.
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