I am very curious about the proof of Algebraic connectivity
The algebraic connectivity of a graph $G$ is the second-smallest eigenvalue of the Laplacian matrix of $G$. This eigenvalue is greater than $0$ if and only if $G$ is a connected graph.This is a corollary to the fact that the number of times $0$ appears as an eigenvalue in the Laplacian is the number of connected components in the graph.
For details : Algebraic connectivity on Wikipedia.
I found this claims very interesting, how exactly the second smallest eigenvalue can be the sign of connectivity of the graph.
Following fact not less interesting,
Denote eigenvalues by $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$, then $\lambda_1=0$.
So far I didn't the proof of Algebraic connectivity. If you have a link, I will appreciate publishing it.