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A Fiedler-like theory for the perturbed Laplacian

Israel Rocha;  Vilmar Trevisan — 2016

Czechoslovak Mathematical Journal

The perturbed Laplacian matrix of a graph G is defined as L D = 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...

Algebraic connectivity of k -connected graphs

Stephen J. Kirkland;  Israel Rocha;  Vilmar Trevisan — 2015

Czechoslovak Mathematical Journal

Let G be a k -connected graph with k ≥ 2 . A hinge is a subset of k vertices whose deletion from G yields a disconnected graph. We consider the algebraic connectivity and Fiedler vectors of such graphs, paying special attention to the signs of the entries in Fiedler vectors corresponding to vertices in a hinge, and to vertices in the connected components at a hinge. The results extend those in Fiedler’s papers Algebraic connectivity of graphs (1973), A property of eigenvectors of nonnegative symmetric...

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