Spectra of M-rooted product of graphs

ABSTRACT

In this paper, we define a new graph operation, namely, M-rooted product of graphs which generalizes the existing rooted product of graphs. We obtain its generalized characteristic polynomial, and as a consequence we deduce the characteristic polynomial of its adjacency, Laplacian and signless Laplacian matrices. Using these results, we derive the L-spectrum of several families of M-rooted product of graphs and deduce several existing results on the spectra of the rooted product of graphs in the literature. As applications, we obtain infinitely many L-cospectral, A-cospectral and $\mathcal{Q}$-cospectral graphs, and construct A-integral graphs and $\mathcal{Q}$-integral graphs.

Keywords:

• Graph products
• adjacency spectrum
• Laplacian spectrum
• signless Laplacian spectrum
• cospectral graphs

2010 Mathematics Subject Classifications:

• 05C50
• 05C76

Acknowledgments

The authors would like to thank the referees for their careful reading and useful comments which have improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author is supported by INSPIRE Fellowship, Departmentof Science and Technology, Government of India [grant number DST/INSPIRE Fellowship/[IF160383] 2017].