# Total behaviour of graph spectrum

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?

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.
• The graph $K_{n,n}$ is not the case, since the wanted spectra is $0$. But I pointed out in the question the connected cases. – Shah Rooz Mar 16 '20 at 4:00