A Laplacian eigenbasis for threshold graphs

Abstract

Let G G be a graph on n n vertices. In this article, we prove that an eigenbasis of the Laplacian matrix of a star graph of order n n is also an eigenbasis of G G if and only if G G is a threshold graph. As an application of this spectral characterization, we show an infinite family of threshold graphs that are weakly Hadamard diagonalizable.