A Note on the Laplacian Eigenvectors of Threshold Graphs

TL;DR

This paper discusses the unique property of threshold graphs having a common Laplacian eigenbasis and provides an alternative proof of this characterization.

Contribution

It offers a new proof for the property that all threshold graphs share a common Laplacian eigenbasis, characterizing threshold graphs.

Findings

All graphs of the same order share a common Laplacian eigenbasis.

This property characterizes threshold graphs.

A different proof of this property is provided.

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.