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., Israel Rocha, Vilmar Trevisan., and Obsahuje seznam literatury
Let $G$ be a $k$-connected graph with $k \ge 2$. 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 matrices and its application to graph theory (1975), and Kirkland and Fallat's paper Perron Components and Algebraic Connectivity for Weighted Graphs (1998).