Canonical form of $C^{\ast }$-algebra of eikonals related to the metric graph

The eikonal algebra $\mathfrak{E}$ of the metric graph $\mathrm{\Omega }$ is an operator $C^{\ast }$–algebra defined by the dynamical system which describes the propagation of waves generated by sources supported in the boundary vertices of $\mathrm{\Omega }$. This paper describes the canonical block form of the algebra $\mathfrak{E}$ of 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 reconstruction of the graph by spectral and dynamical boundary data. Bibliography: 28 items.