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Laplacian spectrum characterization of extensions of vertices of wheel graphs and multi-fan graphs

✍ Scribed by Yuanqing Lin; Jinlong Shu; Yao Meng

Publisher
Elsevier Science
Year
2010
Tongue
English
Weight
294 KB
Volume
60
Category
Article
ISSN
0898-1221

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✦ Synopsis

The graph C n 1 βˆ‡K k is the product of a circuit C n 1 and a clique K k . In this paper, we will prove that it is determined by their Laplacian spectrum except when n 1 = 6. If n 1 = 6, there are several counterexamples. We also prove that the product of s vertex-disjoint paths and a clique (P n 1 βˆͺ P n 2 βˆͺ β€’ β€’ β€’ βˆͺ P ns )βˆ‡K k is also determined by the Laplacian spectrum.

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## Abstract Let __G__ be a simple graph of order __n__ with Laplacian spectrum {Ξ»~__n__~, Ξ»~__n__βˆ’1~, …, Ξ»~1~} where 0=Ξ»~__n__~≀λ~__n__βˆ’1~≀⋅≀λ~1~. If there exists a graph whose Laplacian spectrum is __S__={0, 1, …, __n__βˆ’1}, then we say that __S__ is Laplacian realizable. In 6, Fallat et al. posed