September 20, 2025Open Access

Cubic Edge-Transitive Graphs of Order 2p⁴

Key Points

Connected cubic edge-transitive graphs of order 2p^4 are classified for prime p, establishing their edge-transitive nature.
Using constructions of cubic symmetric bi-Cayley graphs, we show that all classified graphs are symmetric with effective automorphisms.
This study focuses on prime numbers p≥7, emphasizing the interplay between graph symmetry and group structures in classification.
The results reveal significant findings about the automorphism group acting transitively on graph edges in cubic settings.

Abstract

A graph is edge-transitive (s-arc-transitive, respectively) if its full automorphism group Aut\, () acts transitively on the set of edges (the set of s-arcs in for an integer s 0, respectively). A 1-arc-transitive graph is called an arc-transitive graph or a symmetric graph. In this paper, we construct cubic symmetric bi-Cayley graphs over some groups of order p⁴, where p 7 is a prime. Using these constructions, we classify the connected cubic edge-transitive graphs of order 2p⁴ for each prime p and we also show that all these graphs are symmetric.

Cubic Edge-Transitive Graphs of Order 2p⁴

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Abstract

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