A Note on Laplacian Spectrum of Complementary Prisms

Year 2019, Volume: 11, 72 - 80, 30.12.2019

Hande Tunçel Gölpek

Abstract

In this work, the Laplacian spectrum of Complementary Prism graph is considered. The complementary prism operation was introduced by Haynes et al. and denoted by $G\bar{G}$. Some upper and lower bounds obtained using majorization and operator definition of Laplacian. Beside Cardoso et al.'s results in literature about Laplacian spectrum of complementary prisms, an alternative proof about nonzero minimum and maximum Laplacian eigenvalue of complementary prism that contains disconnected components in the underlying graph $G$ or  $\bar{G}$ is provided. Also using this result, the lower and upper bound of nonzero minimum and maximum Laplacian eigenvalue of the complementary prism graph is emphasized.

Keywords

Laplacian Matrix, Eigenvalues, Complementary Prisms

References

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• Fiedler, M., {\em Algebraic connectivity of graphs}, Czech. Math. J., \textbf{23}(1973), 298--305.
• Grone, R., Merris, R., {\em Coalescence, majorization, edge valuations and the Laplacian spectra of graphs}, Linear Multilinear Algebra, \textbf{27}(1990), 139--146.
• Haynes, T.W., Henning, M.A., van der Merwe, L.C., {\em Domination and total domination in complementary prisms}, J. Comb. Optim., \textbf{18}(2009), 23--37.