A graph is totally conformally rigid if and only if it is edge-rigid (every canonical spectral embedding onto a Laplacian eigenspace is edge-isometric), which is equivalent to all edges being pairwise Laplacian-cospectral, enabling a polynomial-time decision algorithm via SDP duality.
Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird
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Total Conformal Rigidity in Graphs
A graph is totally conformally rigid if and only if it is edge-rigid (every canonical spectral embedding onto a Laplacian eigenspace is edge-isometric), which is equivalent to all edges being pairwise Laplacian-cospectral, enabling a polynomial-time decision algorithm via SDP duality.