Spectra of graph operations based on R-graph

Abstract

For a regular graph and an arbitrary graph , we determine the adjacency (respectively, Laplacian and signless Laplacian) spectra of four types of graph operations on and involving the R-graph of , obtained from by adding a new vertex for each edge of and joining each new vertex to the end vertices of the corresponding edge. These results are then used to construct infinitely many pairs of adjacency (respectively, Laplacian and signless Laplacian) cospectral graphs.

Keywords:

• adjacency spectrum
• Laplacian spectrum
• signless Laplacian spectrum
• R-vertex corona
• R-edge corona
• R-vertex neighbourhood corona
• R-edge neighbourhood corona
• cospectral graph

AMS Subject Classifications:

• 05C50

Acknowledgements

The authors thank the referee and the editor for helpful and constructive suggestions and comments.

Additional information

Funding

This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China [grant number 20124407110002]; the National Natural Science Foundation of China [grant number 11071089].