Canonical forms of metric graph eikonal algebra and graph geometry

The algebra of eikonals $\mathfrak{E}$ of a metric graph $\mathrm{\Omega }$ is an operator $C^{\ast }$-algebra determined by dynamical system with boundary control that describes wave propagation on the graph. In this paper, two canonical block forms (algebraic and geometric) of the algebra $\mathfrak{E}$ are provided for an arbitrary connected locally compact graph. These forms determine some metric graphs (frames) ${\mathfrak{F}}^{\mathrm{a}}$ and ${\mathfrak{F}}^{\mathrm{g}}$. Frame ${\mathfrak{F}}^{\mathrm{a}}$ is determined by the boundary inverse data. Frame ${\mathfrak{F}}^{\mathrm{g}}$ is related to graph geometry. A class of ordinary graphs is introduced, whose frames are identical: ${\mathfrak{F}}^{\mathrm{a}}\equiv {\mathfrak{F}}^{\mathrm{g}}$. The results are supposed to be used in the inverse problem that consists in determination of the graph from its boundary inverse data.