Abstract
The perturbed Laplacian matrix of a graph G is defined as DL = 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.
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To the memory of Miroslav Fiedler, whose work and kindness has been an inspiration to many mathematicians all over the world.
This work is part of the doctoral studies of Israel Rocha, who acknowledges the support of CAPES Grant PROBRAL 408/13—Brazil. Vilmar Trevisan is also partially supported by CNPq—Grants 305583/2012-3 and 481551/2012-3.
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Rocha, I., Trevisan, V. A Fiedler-like theory for the perturbed Laplacian. Czech Math J 66, 717–735 (2016). https://doi.org/10.1007/s10587-016-0288-4
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DOI: https://doi.org/10.1007/s10587-016-0288-4