The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of . This eigenvalue is greater than 0 if and only if 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 analyzing the robustness and synchronizability of networks.

Properties

, with the inequality being strict if and only if is connected. However, the algebraic connectivity can be negative for general directed graphs, even if is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity, unless the graph is complete (the algebraic connectivity of a complete graph is its order ). For an undirected connected graph with nonnegative edge weights, vertices, and diameter , and in fact (in a result due to Brendan McKay.Unlike the traditional form of graph connectivity, defined by local configurations whose removal would disconnect the graph, the algebraic connectivity is dependent on the global number of vertices, as well as the way in which vertices are connected. In random graphs, the algebraic connectivity decreases with the number of vertices, and increases with the average degree.

The exact definition of the algebraic connectivity depends on the type of Laplacian used. Fan Chung has developed an extensive theory using a rescaled version of the Laplacian, eliminating the dependence on the number of vertices, so that the bounds are somewhat different.

In models of synchronization on networks, such as the Kuramoto model, the Laplacian matrix arises naturally, so the algebraic connectivity gives an indication of how easily the network will synchronize. Other measures, such as the average distance (characteristic path length) can also be used, and in fact the algebraic connectivity is closely related to the (reciprocal of the) average distance.

The algebraic connectivity also relates to other connectivity attributes, such as the isoperimetric number, which is bounded below by half the algebraic connectivity.

Fiedler vector

The original theory related to algebraic connectivity was produced by Miroslav Fiedler. In his honor the eigenvector associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to partition a graph.

Partitioning a graph using the Fiedler vector

. The negative values are associated with the poorly connected vertex 6, and the neighbouring articulation point, as pictured. The squared values of the components of the Fiedler vector, summing up to one since the vector is normalized, can be interpreted as probabilities of the corresponding data points to be assigned to the sign-based partition.

See also

• Connectivity (graph theory)

• Graph property

References