1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$ – Shah Rooz Mar 16 '20 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. $\endgroup$ – Shah Rooz Mar 16 '20 at 4:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.