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PreprintCC BY 4.0arXiv

Graph transformations and eigenvectors of the Laplacian, signless Laplacian and Adjacency matrices

  • Jean-Guy Caputo
  • Leonardo de Lima
  • Arnaud Knippel
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Abstract

We study graph transformations and their effects on the eigenvectors of three fundamental graph matrices: the adjacency matrix A, the signless Laplacian Q, and the Laplacian L. We extend the Laplacian eigenvector edge and matching principles introduced by Merris to Q and A. We also extend some of the known L graph transformations to Q and A and prove that graphs admitting a common eigenvector x for $A, L$ and Q are such that vertices corresponding to nonzero components of x have same degree. Following our previous work for L, we investigate bivalent graphs for A and Q, that is graphs admitting an eigenvector with all components in {−1, 1}. In particular, for trees, we characterize Q-bivalence and establish that A-bivalence forces all vertex degrees to be odd.

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Cite

BibTeXRIS
BibTeX
@misc{caputo2026graph,
  title = {{Graph transformations and eigenvectors of the Laplacian, signless Laplacian and Adjacency matrices}},
  author = {Jean-Guy Caputo and Leonardo de Lima and Arnaud Knippel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09599},
  eprint = {2610.09599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09599},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Graph transformations and eigenvectors of the Laplacian, signless Laplacian and Adjacency matrices
AU  - Jean-Guy Caputo
AU  - Leonardo de Lima
AU  - Arnaud Knippel
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09599
UR  - https://arxiv.org/abs/2610.09599
AB  - We study graph transformations and their effects on the eigenvectors of three fundamental graph matrices: the adjacency matrix A, the signless Laplacian Q, and the Laplacian L. We extend the Laplacian eigenvector edge and matching principles introduced by Merris to Q and A. We also extend some of the known L graph transformations to Q and A and prove that graphs admitting a common eigenvector x for $A, L$ and Q are such that vertices corresponding to nonzero components of x have same degree. Following our previous work for L, we investigate bivalent graphs for A and Q, that is graphs admitting an eigenvector with all components in {−1, 1}. In particular, for trees, we characterize Q-bivalence and establish that A-bivalence forces all vertex degrees to be odd.
ER  -