Genericity of Simple Eigenvalues for a Metric Graph

Let Γ be a connected finite graph; by V we denote the set of its vertices, and by E we denote the set of its edges. In a contrast with a combinatorial graph, each edge e is considered to be a line segment of length l(e). Sometimes, it is convenient to treat each edge as a pair of oriented edges; then, on an oriented edge, one defines a coordinate xe that runs from 0 to l(e). If −e is the same edge, with the opposite orientation, then x−e = l(e) − xe. If an edge e emanates from a vertex v, we will express it by writing v ≺ e. A good survey of operators on metric graphs and numerous references can be found in [Ku]. A function φ on Γ is a collection of functions φe(x) defined on each edge e. We say that it belongs to L(Γ) if each function φe belongs to L on the corresponding edge; then ||φ|| = ∑