A Note on the Laplacian Eigenvectors of Threshold Graphs

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 $a_1,\dots ,a_r$, such that $a_1⩾2$ and $a_1+a_2+\cdots +a_r=\mathrm{\mid }V(G)\mathrm{\mid }$. 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.