In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

We consider the shape optimization problem \hbox{$\min\big\{\E(\Gamma)\ :\ \Gamma\in\A,\ \H^1(\Gamma)=l\\big\},$}min{ℰ(Γ):Γ ∈ 𝒜,ℋ1(Γ) = l}, where ℋ1 is the one-dimensional Hausdorffmeasure and𝒜is an admissible class of one-dimensional setsconnecting some prescribed set of points \hbox{$\D=\{D_1,\dots,D_k\}\subset\R^d$}𝒟 =  { D1,...,Dk }  ⊂ Rd. The cost functional ℰ(Γ) is theDirichlet energy of Γ defined through the Sobolev functions onΓ vanishing on the pointsDi. We analyze the existence of a solutionin both the families of connected sets and of metric graphs. At the end, several explicitexamples are discussed.

We define sets with finitely ramified cell structure, which are generalizations of post-critically finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.