Extremal Properties of 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. If each edge e is considered as a segment of certain length l(e) > 0 then such a graph is called a metric graph. One can find a good survey and numerous references in [K]. A metric graph with a given combinatorial structure Γ is determined by a vector of edge lengths (l(e)) ∈ R + . We will use the notation G = (Γ, (l(e))). The length of a metric graph, l(G), is the sum of the lengths of all its edges. 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 express it by writing v ≺ e. A function φ on G is a collection of functions φe(x) defined on each edge e. We say that it belongs to L(G) if each function φe belongs to L on the corresponding edge; then ||φ|| = ∑