Spectral Doppelgängers

Different graphs, exactly the same spectral fingerprint.

A graph’s adjacency spectrum can feel like a fingerprint until two different graphs share it. This experiment puts those pairs side by side and shows how the local structure gives them away.

Two non-isomorphic graphs from the working experiment — 01 The selected 12-vertex pair has the same adjacency spectrum. The changed edges and local walk counts let the experiment distinguish the graphs.

What to look for

Look at the changed edges, then the local closed-walk counts. The saved pairs have exact equality certificates for their spectra and a separate obstruction to isomorphism.

Under the hood

Interactive browser experiment with exact Python calculations.

Saved examples, an independent verifier and linked mathematical notes.

The collection includes non-isomorphic graph pairs with matching spectra, exact certificates and a family made by attaching rooted paths.