Laplacian spectra of regular graph transformations

Given a graph GG with vertex set V(G)=VV(G)=V and edge set E(G)=EE(G)=E, let GlGl be the line graph and GcGc the complement of GG. Let G0G0 be the graph with V(G0)=VV(G0)=V and with no edges, G1G1 the complete graph with the vertex set VV, G+=GG+=G and G−=GcG−=Gc. Let B(G)B(G) (Bc(G)Bc(G)) be the graph with the vertex set V∪EV∪E such that (v,e)(v,e) is an edge in B(G)B(G) (resp., in Bc(G)Bc(G)) if and only if v∈Vv∈V, e∈Ee∈E and vertex vv is incident (resp., not incident) to edge ee in GG. Given x,y,z∈{0,1,+,−}x,y,z∈{0,1,+,−}, the xyzxyz-transformation  GxyzGxyzof  GG is the graph with the vertex set V(Gxyz)=V∪EV(Gxyz)=V∪E and the edge set E(Gxyz)=E(Gx)∪E((Gl)y)∪E(W)E(Gxyz)=E(Gx)∪E((Gl)y)∪E(W), where W=B(G)W=B(G) if z=+z=+, W=Bc(G)W=Bc(G) if z=−z=−, WW is the graph with V(W)=V∪EV(W)=V∪E and with no edges if z=0z=0, and WW is the complete bipartite graph with parts VV and EE if z=1z=1. In this paper we obtain the Laplacian characteristic polynomials and some other Laplacian parameters of every xyzxyz-transformation of an rr-regular graph GG in terms of ∣V∣∣V∣, rr, and the Laplacian spectrum of GG.