# Spectral gap

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Mathematical concept

For the physical quantity, see [Spectral gap (physics)](/source/Spectral_gap_(physics)).

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In mathematics, the **spectral gap** is the difference between the [moduli](/source/Absolute_value) of the two largest [eigenvalues](/source/Eigenvalue) of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. The spectral gap gets its name from the *matrix spectrum*, that is, for a matrix, the list of its eigenvalues. It provides insight on diffusion within the graph: corresponding the spectral gap to the smallest non-zero eigenvalue, it is then the mode of the network state that shows the slowest exponential decay over time.

## See also

- [Cheeger constant (graph theory)](/source/Cheeger_constant_(graph_theory))

- [Cheeger constant (Riemannian geometry)](/source/Cheeger_constant)

- [Eigengap](/source/Eigengap)

- [Spectral gap (physics)](/source/Spectral_gap_(physics))

- [Spectral radius](/source/Spectral_radius)

- [Spectral gap conjecture](/source/Spectral_gap_conjecture)

## References

## External links

- ["Impossible-Seeming Surfaces Confirmed Decades After Conjecture"](https://www.quantamagazine.org/impossible-seeming-surfaces-confirmed-decades-after-conjecture-20220602/). *[Quanta Magazine](/source/Quanta_Magazine)*. 2022-06-02.

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