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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?

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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.

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  • $\begingroup$ Thanks @Sam for your answer. I accepted it, but actually, I interested in the cases where the the smallest non negative eigenvalue are not the second largest eigenvalue. $\endgroup$
    – Shahrooz
    Mar 16, 2020 at 3:39
  • $\begingroup$ 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. $\endgroup$
    – Shahrooz
    Mar 16, 2020 at 4:00

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