Geodesic
A Fiedler-like theory for the perturbed Laplacian
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 717-735

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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.
DOI : 10.1007/s10587-016-0288-4
Classification : 05C22, 05C50, 15B57
Keywords: perturbed Laplacian matrix; Fiedler vector; algebraic connectivity; graph partitioning 
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Rocha, Israel; Trevisan, Vilmar. A Fiedler-like theory for the perturbed Laplacian. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 717-735. doi: 10.1007/s10587-016-0288-4

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