Canonical forms of metric graph eikonal algebra and graph geometry

TL;DR

This paper introduces two canonical forms of the eikonal algebra for metric graphs, linking algebraic and geometric representations, and explores their use in inverse problems to determine graph structures from boundary data.

Contribution

It provides algebraic and geometric canonical forms of the eikonal algebra for metric graphs, facilitating inverse problem solutions.

Findings

Two canonical block forms of the eikonal algebra are established.

Frames derived from these forms relate to boundary data and graph geometry.

A class of graphs with identical algebraic and geometric frames is introduced.

Abstract

The algebra of eikonals $\mathfrak E$ of a metric graph $\Omega$ is an operator $C^*$-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) $\mathfrak F^{\,\rm a}$ and $\mathfrak F^{\,\rm g}$. Frame $\mathfrak F^{\,\rm a}$ is determined by the boundary inverse data. Frame $\mathfrak F^{\,\rm g}$ is related to graph geometry. A class of ordinary graphs is introduced, whose frames are identical: $\mathfrak F^{\,\rm a}\equiv\mathfrak F^{\,\rm g}$. The results are supposed to be used in the inverse problem that consists in determination of the graph from its boundary inverse data.