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?



Total behaviour of graph spectrum
The value is infinite. For example, take the friendship graph $F\_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so by considering each of these $G$ in the sum we see that the value must be infinite.


