I want to show a matrix$\begin{bmatrix} L & L_S\\ \alpha L_S&\alpha L \end{bmatrix}$ whose eigenvalues all have non-negative real parts.
where $\alpha$ is an arbrtrary positive scalar. $L$ and $L_S$ are Laplacian-type matrices of an undirected graph $\mathbb{G}$ defined as follows. Let $W\in\mathbb{R}^{m\times m}$ be the adjacency matrix of graph $\mathbb{G}$ such that $W[i,j]=w_{ij}$, with $w_{ij}=1$ if $(i,j)\in\mathcal{E}$ and $w_{ij}=0$ otherwise. Let a diagonal matrix $D\in\mathbb{R}^{m\times m}$ be the degree matrix of graph $\mathbb{G}$, where $D[i,i]=d_i=\sum_{j\in\mathcal{N}_i}w_{ij}$. Then \begin{align} L[i,j]&=\begin{cases} ~~-w_{ij}& \text{for}\quad i\neq j\\ \displaystyle\sum_{j\in\mathcal{N}_i,j\neq i}w_{ij}& \text{for}\quad i=j \end{cases}\\ L_S[i,j]&=\begin{cases} %~~~0& \text{for}\quad j\notin\mathcal{N}_i\\ ~~-s_{ij}=-w_{ij}& \text{for}\quad i< j\\ ~~-s_{ij}=w_{ij}& \text{for}\quad i> j\\ \displaystyle\sum_{j\in\mathcal{N}_i,j\neq i}s_{ij}& \text{for}\quad i=j \end{cases} \end{align}
Example: $L=D-W=\begin{bmatrix}2&-1&-1\\-1&1&0\\-1&0&1\end{bmatrix}$; $L_S=\begin{bmatrix}2&-1&-1\\1&-1&0\\1&0&-1\end{bmatrix}$
Remark: I can prove that $\begin{bmatrix} L &L_S\\ L_S&L \end{bmatrix}$ has eigenvalues with non negative real parts because it is similar to the matrix $\begin{bmatrix} L-L_S &\\ &L+L_S \end{bmatrix}$. By Gershgorin Circle Theorem and the definitions of $L$, $L_S$, both $L+L_S$ and $L-L_S$ have eigenvalues with non-negative real parts (Re$(\lambda)\ge 0$).
However, I'm not able to prove the proposed statement. Here, $L_S$ is not symmetric, which is the main challenge. If you don't consider its diagonal components, it has a skew-symmetric structure.