The Laplacian matrix of weighted threshold graphs

TL;DR

This paper studies the Laplacian matrices of weighted threshold graphs, showing they form a commutative algebra, have common eigenvectors, and that spectra determine graph isomorphism when limited to three or fewer distinct weights.

Contribution

It establishes the algebraic structure of Laplacian matrices for weighted threshold graphs and characterizes spectral properties related to graph isomorphism.

Findings

Laplacian matrices form a commutative algebra.

Eigenvectors are common across the algebra.

Spectral uniqueness for graphs with up to three distinct weights.

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 $w_i$. In this work, we consider the set ${\cal A}_N$ that contains all Laplacian matrices of weighted threshold graphs of order $N$. We show that ${\cal A}_N$ forms a commutative algebra. Using this, we find a common basis of eigenvectors for the matrices in ${\cal A}_N$. It follows that the eigenvalues of each matrix in ${\cal A}_N$ 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.