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Generalized Graph Spectrum

The generalized graph spectrum is the pair consisting of the graph spectrum of a graph and that of its graph complement. A graph is determined by its generalized spectrum if every graph with the same pair of spectra is isomorphic to it.

Xiang (2026) reported that a random graph sampled from G(n,1/2) is determined by its generalized spectrum with probability tending to 1 as n->infty. This is a weaker statement than the Haemers conjecture, which uses only the spectrum of the graph itself. Xiang (2026) describes the work as a collaboration with Claude Opus 5.5.

See also

Determined by Spectrum, Graph Complement, Graph Spectrum, Haemers Conjecture

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References

Xiang, Z. "Almost All Graphs Are Determined by Their Generalized Spectrum." 8 Oct 2026. https://arxiv.org/abs/2610.11047.

Cite this as:

Weisstein, Eric W. "Generalized Graph Spectrum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedGraphSpectrum.html

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