Reducing the adjacency matrix of a tree.
Fricke, Gerd H.; Hedetniemi, Stephen T.; Jacobs, David P.; Trevisan, Vilmar — 1996
ELA. The Electronic Journal of Linear Algebra [electronic only]
Reducing the adjacency matrix of a tree.
A Fiedler-like theory for the perturbed Laplacian
The perturbed Laplacian matrix of a graph is defined as , where is any diagonal matrix and is a weighted adjacency matrix of . 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 -connected graphs
Let be a -connected graph with . A hinge is a subset of vertices whose deletion from 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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