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Laplacian state transfer in coronas. (English) Zbl 1346.05158

Summary: We prove that the corona product of two graphs has no Laplacian perfect state transfer whenever the first graph has at least two vertices. This complements a result of G. Coutinho and H. Liu [SIAM J. Discrete Math. 29, No. 4, 2179–2188 (2015; Zbl 1327.05202)] who showed that no tree of size greater than two has Laplacian perfect state transfer. In contrast, we prove that the corona product of two graphs exhibits Laplacian pretty good state transfer, under some mild conditions. This provides the first known examples of families of graphs with Laplacian pretty good state transfer. Our result extends the work of X. Fan and C. Godsil [Linear Algebra Appl. 438, No. 5, 2346–2358 (2013; Zbl 1258.05069)] on double stars to the Laplacian setting. Moreover, we also show that the corona product of any cocktail party graph with a single vertex graph has Laplacian pretty good state transfer, even though odd cocktail party graphs have no perfect state transfer.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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