Adjacency polynomials of digraph transformations

Let A(λ,D)A(λ,D) be the adjacency characteristic polynomial of a digraph DD. In the paper Deng and Kelmans (2013) the so-called (xyz)(xyz)-transformation DxyzDxyz of a simple digraph DD was considered, where x,y,z∈{0,1,+,−}x,y,z∈{0,1,+,−}, and the formulas of A(λ,Dxyz)A(λ,Dxyz) were obtained for every rr-regular digraph DD in terms of rr, the number of vertices of DD, and A(λ,D)A(λ,D). In this paper we define the so-called (xyab)(xyab)-transformation DxyabDxyab of a simple digraph DD, where x,y,a,b∈{0,1,+,−}x,y,a,b∈{0,1,+,−}. This notion generalizes the previous notion of the (xyz)(xyz)-transformation DxyzDxyz, namely, Dxyab=DxyzDxyab=Dxyz if and only if a=b=za=b=z. We extend our previous results on A(λ,Dxyz)A(λ,Dxyz) to the (xyab)(xyab)-transformation DxyabDxyab by obtaining the formulas of A(λ,Dxyab)A(λ,Dxyab), where x,y,a,b∈{0,1,+,−}x,y,a,b∈{0,1,+,−} and a≠ba≠b, for every simple rr-regular digraph DD in terms of rr, the number of vertices of DD, and A(λ,D)A(λ,D). We also use (xyab)(xyab)-transformations to describe various constructions providing infinitely many examples of adjacency cospectral non-isomorphic digraphs.