Spectral Properties of the Laplacian
The Laplacian matrix of a graph, as introduced earlier, encodes both the degree of each node and the adjacency structure of the graph. The spectrum of the Laplacian refers to the set of its eigenvalues, which are always real and nonnegative due to the Laplacian's symmetric and positive semidefinite nature. These eigenvalues, often ordered as λ0≤λ1≤...≤λn−1, reveal deep insights about the graph's connectivity and structure.
Theorem (Connected Components and Zero Eigenvalues):
The multiplicity of the zero eigenvalue of the Laplacian matrix of a graph is equal to the number of connected components in the graph. In other words, if the Laplacian has k zero eigenvalues, the graph consists of k disjoint connected parts.
The Laplacian's eigenvalues measure how "easy" it is to separate a graph into disconnected pieces. If a graph is fully connected, there is only one way to have zero "flow" across edges — by assigning the same value to every node. If the graph splits into multiple disconnected parts, each part can independently have its own constant value, leading to more zero eigenvalues.
The number of linearly independent eigenvectors associated with the eigenvalue zero (the nullity of the Laplacian) equals the number of connected components. The smallest nonzero eigenvalue, called the algebraic connectivity or Fiedler value, quantifies how well the graph is connected: the larger this value, the more difficult it is to separate the graph into disconnected pieces.
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The Laplacian matrix of a graph, as introduced earlier, encodes both the degree of each node and the adjacency structure of the graph. The spectrum of the Laplacian refers to the set of its eigenvalues, which are always real and nonnegative due to the Laplacian's symmetric and positive semidefinite nature. These eigenvalues, often ordered as λ0≤λ1≤...≤λn−1, reveal deep insights about the graph's connectivity and structure.
Theorem (Connected Components and Zero Eigenvalues):
The multiplicity of the zero eigenvalue of the Laplacian matrix of a graph is equal to the number of connected components in the graph. In other words, if the Laplacian has k zero eigenvalues, the graph consists of k disjoint connected parts.
The Laplacian's eigenvalues measure how "easy" it is to separate a graph into disconnected pieces. If a graph is fully connected, there is only one way to have zero "flow" across edges — by assigning the same value to every node. If the graph splits into multiple disconnected parts, each part can independently have its own constant value, leading to more zero eigenvalues.
The number of linearly independent eigenvectors associated with the eigenvalue zero (the nullity of the Laplacian) equals the number of connected components. The smallest nonzero eigenvalue, called the algebraic connectivity or Fiedler value, quantifies how well the graph is connected: the larger this value, the more difficult it is to separate the graph into disconnected pieces.
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