Nodal Domain Theorems for Special Graph Classes

In Sect. 3.6 we have seen that the upper bound for the number of (strong or weak) nodal domains that is given by the discrete nodal domain theorem cannot be improved without further restrictions. On the other hand, we have seen that there exist graphs where this bound is not sharp. In general it is unknown, whether this upper bound is sharp for an arbitrary graph. The situation is similar for the (trivial) lower bound in Thm. 3.33. Furthermore, no generalmethod is known to construct an eigenfunction of a given eigenvalue λk that maximizes or minimizes the number of (strong or weak) nodal domains. In this chapter we take a closer look to the situation for trees, cographs, and product graphs (in particular to the Boolean hypercube), where it is possible to derive improved upper and lower bounds.