Canonical forms of metric graph eikonal algebra and graph geometry

The algebra of eikonals $\mathfrak E$ of a metric graph $\Omega$ is an operator $C^*$-algebra defined by a dynamical system with boundary control describing wave propagation. In this paper, two canonical block forms of the algebra $\mathfrak E$ are described for an arbitrary connected locally compact graph – algebraic and geometric. These forms define some metric graphs (frames) $\mathfrak F^{ \rm a}$ and $\mathfrak F^{ \rm g}$. The frame $\mathfrak F^{ \rm a}$ is defined by the boundary data of inverse problems. Frame $\mathfrak F^{ \rm g}$ is related to graph geometry. A class is being introduced of ordinary graphs, whose frames are identical: $\mathfrak F^{ \rm a}\equiv\mathfrak F^{ \rm g}$. The results are supposed to be used in the inverse problem, which consists in reconstructing a graph from boundary inverse data.