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

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• 作者：Nair Abreu^a ; ^{nairabreunovoa@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Domingos M. Cardoso^b ; ^c ; ^{dcardoso@ua.pt" class="auth_mail" title="E-mail the corresponding author} ; Paula Carvalho^b ; ^c ; ^{paula.carvalho@ua.pt" class="auth_mail" title="E-mail the corresponding author} ; Cybele T.M. Vinagre^d ; ^{cybl@vm.uff.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Graph spectra ; Graph operations ; Lexicographic product of graphs
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 January 2017
• 年：2017
• 卷：340
• 期：1
• 页码：3235-3244
• 全文大小：976 K

文摘

Consider two graphs e6fe84cc5c7130" title="Click to view the MathML source">G$G$ and 961893e35e99b6a2c50005da088" title="Click to view the MathML source">H$H$. Let 8a56daf" title="Click to view the MathML source">H^k[G]$H^k\left[G\right]$ be the lexicographic product of afdcdcafed67" title="Click to view the MathML source">H^k$H^k$ and e6fe84cc5c7130" title="Click to view the MathML source">G$G$, where afdcdcafed67" title="Click to view the MathML source">H^k$H^k$ is the lexicographic product of the graph 961893e35e99b6a2c50005da088" title="Click to view the MathML source">H$H$ by itself afe" title="Click to view the MathML source">k$k$ times. In this paper, we determine the spectrum of 8a56daf" title="Click to view the MathML source">H^k[G]$H^k\left[G\right]$ and afdcdcafed67" title="Click to view the MathML source">H^k$H^k$ when e6fe84cc5c7130" title="Click to view the MathML source">G$G$ and 961893e35e99b6a2c50005da088" title="Click to view the MathML source">H$H$ are regular and the Laplacian spectrum of 8a56daf" title="Click to view the MathML source">H^k[G]$H^k\left[G\right]$ and afdcdcafed67" title="Click to view the MathML source">H^k$H^k$ for e6fe84cc5c7130" title="Click to view the MathML source">G$G$ and 961893e35e99b6a2c50005da088" title="Click to view the MathML source">H$H$ 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.