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Spectra of M-edge rooted product of graphs

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Abstract

In this paper, we define a graph operation, namely, M-edge rooted product of graphs. This generalizes the existing graph operation called graphs with edge pockets. Also we introduce a matrix invariant, namely, coronal of a matrix constrained by the index sets. We compute this value for some class of matrices with respect to some index sets. We obtain the generalized characteristic polynomial of the graph obtained by M-edge rooted product with a help of this invariant. Consequently, we deduce the characteristic polynomial of the adjacency matrix, the Laplacian matrix and the signless Laplacian matrix of this graph. Using these results, we derive the L-spectrum of several families of M-edge rooted product of graphs and deduce several existing results on the spectra of graphs with edge pockets in the literature. As applications, we obtain infinitely many L-cospectral graphs and construct A-integral graphs, L-integral graphs.

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Acknowledgements

The authors express their sincere thanks to the referee for his/her careful reading and useful comments which have improved the paper. The first author is supported by INSPIRE Fellowship, Department of Science and Technology, Government of India under the Grant No. DST/INSPIRE Fellowship/[IF160383] 2017.

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Authors and Affiliations

  1. Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Tamil Nadu, 624 302, India

    R. Pavithra & R. Rajkumar

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  1. R. Pavithra
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  2. R. Rajkumar
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Correspondence to R. Rajkumar.

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Communicated by Ravindra B Bapat, Prof.

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Pavithra, R., Rajkumar, R. Spectra of M-edge rooted product of graphs. Indian J Pure Appl Math 52, 1235–1255 (2021). https://doi.org/10.1007/s13226-021-00027-6

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  • DOI: https://doi.org/10.1007/s13226-021-00027-6

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