A Classification of 4-Degree Tri-Cayley Graphs Over a Group of Order 𝑝q

Symmetry properties are of vital importance for graphs. Meanwhile, graphs with the vertex transitivity are a class of highly symmetrical graphs. A graph 𝛷 is said to be a tri-Cayley graph over a group 𝐻 if it has a semi-regular automorphism group which acts on the vertex set with three orbits of equal length and is isomorphic to 𝐻. In this paper, the vertex transitivity, edge transitivity and arc transitivity of the 4-degree 0-type and 2-type tri-Cayley graphs over a group ℤ𝑝𝑞 are discussed and give the automorphism group of the corresponding vertex transitive graph.