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The Laplacian matrix of weighted threshold graphs
Ke,Yingyue Haemers,W. H. Mieghem,Piet Van
Ke,Yingyue
Haemers,W. H.
Mieghem,Piet Van
Abstract
Threshold graphs are generated from one node by repeatedly adding a node that links to all existing nodes or adding a node without links. In the weighted threshold graph, we add a new node in step i, which is linked to all existing nodes by a link of weight wi. In this work, we consider the set AN that contains all Laplacian matrices of weighted threshold graphs of order N. We show that AN forms a commutative algebra. Using this, we find a common basis of eigenvectors for thematrices in AN. It follows that the eigenvalues of each matrix in AN can be represented as a linear transformation of the link weights. In addition, we prove that, if there are just three or fewer different weights, two weighted threshold graphs with the same Laplacian spectrum must be isomorphic.
Description
Date
2025-10-04
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Research Projects
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Keywords
Threshold graphs, Laplacian matrix, commutative algebra, cospectral graphs
Citation
Ke, Y, Haemers, W H & Mieghem, P V 2025, 'The Laplacian matrix of weighted threshold graphs', Electronic Journal of Linear Algebra, vol. 41, pp. 529-537. https://doi.org/10.13001/ela.2025.9653