Canonical form of the $C^*$-algebra of eikonals related to a metric graph

The eikonal algebra $\mathfrak E$ of a metric graph $\Omega$ is an operator $C^*$-algebra defined by the dynamical system which describes the propagation of waves generated by sources supported at the boundary vertices of $\Omega$. This paper describes the canonical block form of the algebra $\mathfrak E$ for an arbitrary compact connected metric graph. Passing to this form is equivalent to constructing a functional model which realizes $\mathfrak E$ as an algebra of continuous matrix-valued functions on its spectrum $\widehat{\mathfrak{E}}$. The results are intended to be used in the inverse problem of recovering the graph from spectral and dynamical boundary data.