Laplacian spectra of regular graph transformations

Abstract

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 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-transformationGxyzofG 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.