The signless Laplacian state transfer in coronas

Abstract

For two graphs G and H, the corona product $G\circ H$ is the graph obtained by taking one copy of G and $|V_G|$ copies of H, and joining the ith vertex of G with every vertex of the ith copy of H. In this paper, we study the state transfer of corona relative to the signless Laplacian matrix. We explore some conditions that guarantee the signless Laplacian perfect state transfer in $G\circ H$. We prove that $G\circ K_m$ has no signless Laplacian perfect state transfer for some special m. We also show that $K_2\circ H$ has pretty good state transfer but no perfect state transfer relative to the signless Laplacian matrix for a regular graph H. Furthermore, we show that $\overline{n{K}_2}\circ K_1$ has signless Laplacian pretty good state transfer, where $\overline{n{K}_2}$ is the cocktail party graph.

COMMUNICATED BY:

• C.-K. Li

Keywords:

• Quantum walk
• signless Laplacian
• spectrum
• corona
• state transfer

AMS classifications:

• 05C50
• 05C12
• 15A18
• 81P45

Acknowledgments

We would like to thank the anonymous referee for careful reading of our manuscript and for invaluable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was in part supported by the National Natural Science Foundation of China (Nos. 11801521, 11671053).