On graphs with a few distinct reciprocal distance Laplacian eigenvalues

Milica Anđelić; Saleem Khan; S. Pirzada; Milica Anđelić; Saleem Khan; S. Pirzada

doi:10.3934/math.20231485

AIMS Mathematics

2023, Volume 8, Issue 12: 29008-29016. doi: 10.3934/math.20231485

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Research article

On graphs with a few distinct reciprocal distance Laplacian eigenvalues

1.
Department of Mathematics, Kuwait University, Al-Shadadiyah, Kuwait
2.
Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India

Received: 17 July 2023 Revised: 17 September 2023 Accepted: 02 October 2023 Published: 25 October 2023
MSC : 05C50, 05C12, 15A18

For a $ \nu $-vertex connected graph $ \Gamma $, we consider the reciprocal distance Laplacian matrix defined as $ RD^L(\Gamma) = RT(\Gamma)-RD(\Gamma) $, i.e., $ RD^L(\Gamma) $ is the difference between the diagonal matrix of the reciprocal distance degrees $ RT(\Gamma) $ and the Harary matrix $ RD(\Gamma) $. In this article, we determine the graphs with exactly two distinct reciprocal distance Laplacian eigenvalues.We completely characterize the graph classes with a $ RD^L $ eigenvalue of multiplicity $ \nu-2 $. Moreover, we characterize families of graphs with reciprocal distance Laplacian eigenvalue whose multiplicity is $ \nu-3 $.
- Harrary matrix,
- reciprocal distance Laplacian matrix,
- reciprocal distance Laplacian eigenvalues
Citation: Milica Anđelić, Saleem Khan, S. Pirzada. On graphs with a few distinct reciprocal distance Laplacian eigenvalues[J]. AIMS Mathematics, 2023, 8(12): 29008-29016. doi: 10.3934/math.20231485

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Abstract

For a $ \nu $-vertex connected graph $ \Gamma $, we consider the reciprocal distance Laplacian matrix defined as $ RD^L(\Gamma) = RT(\Gamma)-RD(\Gamma) $, i.e., $ RD^L(\Gamma) $ is the difference between the diagonal matrix of the reciprocal distance degrees $ RT(\Gamma) $ and the Harary matrix $ RD(\Gamma) $. In this article, we determine the graphs with exactly two distinct reciprocal distance Laplacian eigenvalues.We completely characterize the graph classes with a $ RD^L $ eigenvalue of multiplicity $ \nu-2 $. Moreover, we characterize families of graphs with reciprocal distance Laplacian eigenvalue whose multiplicity is $ \nu-3 $.

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