Nodal domains for discrete Laplacians on graphs: morphology, percolation and isospectrality

One of the most outstanding discoveries in the nineteen century was Chladni's work on the vibration of plates, in which he discovered the intricate patterns of nodal lines - the lines where the vibration amplitude vanishes. The areas bounded by the nodal lines are the nodal domains, whose counts and morphologies were the subject of many studies till this very day. However, only recently was it realized, that the complexity and richness of forms displayed by nodal domains increase dramatically if the domains of study were of general shapes (and not the squares and discs used originally). Further surprises followed when detailed analysis of the nodal morphologies was attempted. Indeed, the best description of this phenomenon is taken from percolation theory - a statistical theory which a priory has nothing to do with wave dynamics! Another surprising discovery emerged in the attempts to understand the way by which the number of nodal domains depend on the frequency of the vibration. It turns out that in some cases, the sequence of this numbers (the nodal count) can be used to deduce the geometry of the vibrating domain, and in some cases even to specify it completely. These results, and several others of more technical nature have drawn the attention of both mathematical and physical communities to the study of nodal domains.In the present work we would like to contribute to this study by an analysis of a class of models which are simpler to investigate, yet preserve much of the complexity of the original problem. These are graphs which can be thought of as the representations of the vibrating membranes by a network of intricately connected wires. Indeed, preliminary studies show that many of the characteristics of nodal domains are expressed also in their graph models. Graphs, however are simpler to handle, and by establishing rigorous results on graphs, we shell shed light on the subject, and acquire the skill and the insight which will be useful in addressing the problem in its general setting.