Perron components and algebraic connectivity for weighted graphs

The algebraic connectivity of a connected graph is the second-smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigenvalue. In this paper, we introduce the notion of a perron component at a vertex in a weighted graph, and show how the structure of the eigenvectors associated with the algebraic connectivity can be understood in terms of perron components. This leads to some strengthening of Fiedler's original result, gives some insights into weighted graphs under perturbation, and allows for a discussion of weighted graphs exhibiting tree-like structure.