Transportation in graphs and the admittance spectrum

The admittance spectrum of the underlying graph is used in order to describe the behaviour of a liquid flowing through a system of communicating pipes. The performance of this process enables us to define the ‘permeability’ of a graph, the ‘well-connectedness’ of pairs of vertices and the ‘good position’ of vertices in terms of the second eigenvalue of the admittance matrix and its corresponding eigenvectors. Furthermore, bounds are given for the decrease of the second eigenvalue caused by the insertion of an edge into a graph.