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arXiv · 2605.03645

A Note on the Laplacian Eigenvectors of Threshold Graphs

James L. Borg·Irene Sciriha·Zoia Sherman
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Abstract

Threshold graphs are graphs that can be characterized in a number of different ways. For example, they are graphs that are $P_4,\ C_4,\ 2K_2$--free. They may also be characterized by a finite sequence of positive integers $a_1, \ldots, a_r$, such that $a_1\geqslant 2$ and $a_1 + a_2 + \cdots + a_r = |V(G)|$. Threshold graphs have the remarkable property that all graphs of the same order share a common integer Laplacian eigenbasis. This property characterizes threshold graphs. This result was proved in \cite{MachareteDelVecchio}. We give a different proof of the same result.

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James L. Borg, Irene Sciriha, Zoia Sherman. 2026-05-05. A Note on the Laplacian Eigenvectors of Threshold Graphs. https://arxiv.org/abs/2605.03645

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