Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs GG and HH. Let Hk[G]Hk[G] be the lexicographic product of HkHk and GG, where HkHk is the lexicographic product of the graph HH by itself kk times. In this paper, we determine the spectrum of Hk[G]Hk[G] and HkHk when GG and HH are regular and the Laplacian spectrum of Hk[G]Hk[G] and HkHk for GG and HH arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.