Automorphism Group of Some A-join of Graphs

For a simple graph A, the A-join Γ of a set of complete or empty graphs {Aa}a∈V(A) is an another simple graph with the vertex and edge sets {(a;b) │ a ∈ V(A) b ∈V(Aa)} and {(a;b)(a′;b′) │ aa′ ∈ E(A) or else a = a′ bb′ ∈ E(Aa)}, respectively. Th is graph is called reduced if NA(a) \ {b} = NA(b) \ {a}, a;b ∈ V(A) and a ≠ b, implies that (i) if ab ∉ E(A) then at least Aa or Ab are non-empty; (ii) if ab ∈ E(A) then at least one of Aa or Ab are not complete. In this talk, we report our recent results on the automorphism group of reduced completeempty A-join of graphs.