Algebraic connectivity

The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue) 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. The magnitude of this value reflects how well connected the overall graph is. It has been used in analysing the robustness and synchronizability of networks.

The algebraic connectivity of a graph G is greater than 0 if and only if G is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity of the graph. If the number of vertices of a connected graph is n and the diameter is D, the algebraic connectivity is known to be bounded below by ${\displaystyle {\frac {1}{nD}}}$, and in fact (in a result due to Brendan McKay) by ${\displaystyle {\frac {4}{nD}}}$. For the example shown above, 4/18 = 0.222 ≤ 0.722 ≤ 1.

...
Wikipedia