Laplacian spectra of regular graph transformations

Given a graph G with vertex set V (G) = V and edge set E(G) = E, let Gl be the line graph and Gc the complement of G. Let G0 be the graph with V (G0) = V and with no edges, G1 the complete graph with the vertex set V , G+ = G and G− = Gc. Let B(G) (Bc(G)) be the graph with the vertex set V ∪E and such that (v, e) is an edge in B(G) (resp., in Bc(G)) if and only if v ∈ V , e ∈ E and vertex v is incident (resp., not incident) to edge e in G. Given x, y, z ∈ {0, 1,+,−}, the xyz-transformation Gxyz of G is the graph with the vertex set V (Gxyz) = V ∪ E and the edge set E(Gxyz) = E(Gx)∪E((Gl)y)∪E(W ), where W = B(G) if z = +, W = Bc(G) if z = −, W is the graph with V (W ) = V ∪ E and with no edges if z = 0, and W is the complete bipartite graph with parts V and E if z = 1. In this paper we obtain the Laplacian characteristic polynomials and some other Laplacian parameters of every xyz-transformation of an r-regular graph G in terms of |V |, r, and the Laplacian spectrum of G.