Spectra of subdivision-vertex and subdivision-edge neighbourhood coronae

Let G = (V (G), E(G)) be a graph with vertex set V (G) and edge set E(G). The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let G(1) and G(2) be two vertex disjoint graphs. The subdivision-vertex neighbourhood corona of G(1) and G(2), denoted by G(1) boxed dot G(2), is the graph obtained from S(G(1)) and vertical bar V(G(1))vertical bar copies of G(2), all vertex disjoint, and joining the neighbours of the ith vertex of V(G1) to every vertex in the ith copy of G(2). The subdivision-edge neighbourhood corona of G(1) and G(2), denoted by G(1) boxed minus G(2), is the graph obtained from S(G(1)) and vertical bar I(G(1))vertical bar copies of G(2), all vertex disjoint, and joining the neighbours of the ith vertex of I(G(1)) to every vertex in the ith copy of G(2), where I(G(1)) is the set of inserted vertices of S(G(1)). In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of G(1) boxed dot G(2) (respectively, G(1) boxed minus G(2)) in terms of the corresponding spectra of G(1) and G(2). As applications, these results enable us to construct infinitely many pairs of cospectral graphs, and using the results on the Laplacian spectra of subdivision-vertex neighbourhood coronae, new families of expander graphs are constructed from known ones. (C) 2013 Elsevier Inc. All rights reserved.