Let $G=(V,E)$ be the disjoint union of two graphs$G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. That is, $V_1$ and $V_2$, $V=V_1 \cup V_2$ and $E=E_1\cup E_2$. We want to show that $$\theta(G) = \theta(G_1) + \theta(G_2) $$ Where we have multiple definitions for $\theta(G)$:
- $\min\{ t \mid C-J\succeq 0$ , $C_{ii} = t$ $\forall i\in V$, $C_{ij}=0$ for $\{i,j\}\in \bar{E}\}$ (With $J$ matrix of all ones)
- $\max\{ \Sigma_{i\in V} Y_{ii} \mid Y\succeq 0$, $Y_{00} =1, Y_{ij}=0\text{ for }\{i,j\}\in E,\:Y_{0i}=Y_{ii}\}$
Who can help us? I think you want to prove it in a $\theta(G) \leq \theta(G_1) + \theta(G_2) $ and $\theta(G) \geq \theta(G_1) + \theta(G_2) $ structure.