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Keywords:
perturbed Laplacian matrix; Fiedler vector; algebraic connectivity; graph partitioning
perturbed Laplacian matrix; Fiedler vector; algebraic connectivity; graph partitioning
Summary:
The perturbed Laplacian matrix of a graph $G$ is defined as $L^{\mkern -15muD}=D-A$, where $D$ is any diagonal matrix and $A$ is a weighted adjacency matrix of $G$. We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore, we use the notion of Perron component for the perturbed Laplacian matrix of a graph and show how its second smallest eigenvalue can be characterized using this definition.
References:
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