I will give a brief introduction to quantum graphs and the nodal count sequence. I will then present the known bounds on the nodal count and define the nodal surplus. Following these I will state a theorem about the convergence of the average nodal surplus to a distribution which we call the nodal count distribution, and then state a theorem that show an interesting relation between the topology of a graph and its nodal count distribution.

This work is based on a joint work with Ram Band and Gregory Berkolaiko.