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. The magnitude of this value reflects how well connected the overall graph is, and has been used in analysing the synchronizability of networks.

Properties

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.J.L. Gross and J. Yellen. Handbook of Graph Theory, CRC Press, 2004, page 314. 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 1/nD,J.L. Gross and J. Yellen. Handbook of Graph Theory, CRC Press, 2004, page 571. and in fact (in a result due to Brendan McKay) by 4/nD.Bojan Mohar, , in Graph Theory, Combinatorics, and Applications, Vol. 2, Ed. Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk, Wiley, 1991, pp. 871–898. For the example shown above, for example,...
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