Spectra of the neighbourhood corona of two graphs

Abstract

Given simple graphs and , the neighbourhood corona of and , denoted , is the graph obtained by taking one copy of and copies of , and joining the neighbours of the th vertex of to every vertex in the th copy of . In this paper we determine the adjacency spectrum of for arbitrary and , and the Laplacian spectrum and signless Laplacian spectrum of for regular and arbitrary , in terms of the corresponding spectrum of and . The results on the adjacency and signless Laplacian spectra enable us to construct new pairs of adjacency cospectral and signless Laplacian cospectral graphs. As applications of the results on the Laplacian spectra, we give constructions of new families of expander graphs from known ones by using neighbourhood coronae.