In the past decade, there has been extensive research on the nonlinear Schrödinger equation (NLS) on metric graphs, driven by both the physical and mathematical communities. Metric graphs, in essence, are one-dimensional objects that can model network-like structures. The first goal of this talk, particularly for those who are new to metric graphs, is to provide an introduction to these structures and present the appropriate functional framework for studying the NLS equation on them.

It is well-established that both the metric (size) and topological (shape) properties of the graphs can impact the existence of solutions to the NLS. As a result, no general theory currently exists for analyzing the NLS equation on metric graphs. A common approach is to focus on specific classes of graphs. In this talk, we focus on two such graphs: the $\mathcal{T}$-graph and tadpole graphs. We then discuss, using techniques from the theory of ordinary differential equations (specifically, parts of the period function), how to approach questions related to the existence, uniqueness, and multiplicity of positive solutions on these graphs.

Time permitting, we will demonstrate how this careful analysis leads to a series of existence and uniqueness/multiplicity results for a class of graphs known as single-knot graphs.

This talk is based on joint work with Simão Correia and Hugo Tavares.