Let $\mathcal{G}$ be the set of all finite connected simple graphs minus the complete graphs. For any $G\in \mathcal{G}$, let $\lambda_{\geq0}(G)$ denotes the smallest positive adjacency eigenvalue of the graph $G$.

Let $$\tau_{\mathcal{G}}=\sum_{G\in \mathcal{G}}{\lambda_{\geq 0}(G)}.$$

Is it true that $\tau_{\mathcal{G}}$ is finite?