Cubic edge-transitive bi-$p$-metacirculant

TL;DR

This paper characterizes the automorphism groups of connected cubic edge-transitive bi-p-metacirculants, revealing their existence only for p=3, and classifies certain bi-Cayley graphs, leading to the construction of an infinite family of cubic semisymmetric graphs.

Contribution

It provides a complete characterization of automorphism groups of these graphs and classifies them over specific groups, introducing new infinite families of semisymmetric graphs.

Findings

Connected cubic edge-transitive bi-p-metacirculants exist only for p=3.

Classified bi-Cayley graphs over inner-abelian metacyclic 3-groups.

Constructed the first infinite family of cubic semisymmetric graphs of order twice a 3-power.

Abstract

A graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime p, we call a bi-Cayley graph over a metacyclic p-group a bi-p-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-p-metacirculant is characterized for an odd prime p, and the result reveals that a connected cubic edge-transitive bi-p-metacirculant exists only when p=3. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic 3-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a 3-power.