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The algebraic connectivity of a finite simple graph
Definition
Let be a finite simple graph with , and let be its Laplacian matrix. Because is real symmetric and positive semidefinite, its eigenvalues are real and nonnegative (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis, The Laplacian is positive semidefinite and sends the all-ones vector to zero). Moreover, is an eigenvalue by The multiplicity of the Laplacian eigenvalue equals the number of connected components, so the eigenvalues may be listed in weakly increasing order as
The second eigenvalue is the algebraic connectivity of .
This quantity is defined only for graphs with at least two vertices, because a one-vertex graph has only one Laplacian eigenvalue.
Depends on
Used by
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Sources
- O. Pikhurko, Algebraic Methods in Combinatorics, Section 14.2 (standard reference, not scraped)