Is it true that $L+L^T$ has at most one negative eigenvalue?

Question

Suppose $L$ is a non-symmetric Laplacian matrix, then $L+L^T$ is symmetric. Is it true that $L+L^T$ has at most one negative eigenvalue? I try many examples using Matlab numerically to find out that there is at most one negative eigenvalues, but I am not sure it is true or not.

Answer 1

This is not true. If $L+L^T$ has a negative eigenvalue (such as when $L=\pmatrix{1&-1\\ 0&0}$), then $L'+L'^T$ can have as many negative eigenvalues as you like, when $L'=L\oplus L\oplus\cdots\oplus L$ is a direct sum of multiple copies of $L$.