What does it mean if the two smallest eigenvalues of the Laplacian matrix of a graph are equal to zero? 



If the two smallest eigenvalues of the Laplacian matrix of a network are equal to zero, then does it mean that the network is not connected?
If by "network" you mean "graph", then yes, the eigenvalue 0 having multiplicity at least two means that the graph has at least two connected components. This depends on the fact that the characteristic vector of each connected component is always an eigenvector for the eigenvalue 0.


If by network you mean something different (an oriented graph? perhaps with some weights and/or capacities?), then you should first explain what you mean by "Laplacian matrix" of it.


