Geodesic
Canonical forms of metric graph eikonal algebra and graph geometry
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 50, Tome 519 (2022), pp. 35-66 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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M. I. Belishev; A. V. Kaplun. Canonical forms of metric graph eikonal algebra and graph geometry. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 50, Tome 519 (2022), pp. 35-66. http://geodesic.mathdoc.fr/item/ZNSL_2022_519_a2/

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