文摘
Consider two graphs 82&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=eee5701ac9c753f66fe6fe84cc5c7130" title="Click to view the MathML source">G and 82&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=da62f961893e35e99b6a2c50005da088" title="Click to view the MathML source">H. Let 82&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=2d5758bd7ec7237fdff7c4c888a56daf" title="Click to view the MathML source">Hk[G] be the lexicographic product of 82&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=10a23dbf68c5c1bd2854afdcdcafed67" title="Click to view the MathML source">Hk and 82&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=eee5701ac9c753f66fe6fe84cc5c7130" title="Click to view the MathML source">G, where 82&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=10a23dbf68c5c1bd2854afdcdcafed67" title="Click to view the MathML source">Hk is the lexicographic product of the graph 82&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=da62f961893e35e99b6a2c50005da088" title="Click to view the MathML source">H by itself 82&_mathId=si11.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=6898e1c6a70851437d3533143284fafe" title="Click to view the MathML source">k times. In this paper, we determine the spectrum of 82&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=2d5758bd7ec7237fdff7c4c888a56daf" title="Click to view the MathML source">Hk[G] and 82&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=10a23dbf68c5c1bd2854afdcdcafed67" title="Click to view the MathML source">Hk when 82&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=eee5701ac9c753f66fe6fe84cc5c7130" title="Click to view the MathML source">G and 82&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=da62f961893e35e99b6a2c50005da088" title="Click to view the MathML source">H are regular and the Laplacian spectrum of 82&_mathId=si6.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=2d5758bd7ec7237fdff7c4c888a56daf" title="Click to view the MathML source">Hk[G] and 82&_mathId=si7.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=10a23dbf68c5c1bd2854afdcdcafed67" title="Click to view the MathML source">Hk for 82&_mathId=si4.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=eee5701ac9c753f66fe6fe84cc5c7130" title="Click to view the MathML source">G and 82&_mathId=si5.gif&_user=111111111&_pii=S0012365X16302382&_rdoc=1&_issn=0012365X&md5=da62f961893e35e99b6a2c50005da088" title="Click to view the MathML source">H 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.