In general, it is difficult to determine if a graph is determined by spectrum. van Dam and Haemers (van Dam and Haemers 2002, Haemers 2016) noted
that it is conceivable that *almost all* graphs have this property, i.e., that
the fraction of graphs on vertices with a cospectral mate tends to zero as tends to infinity, an assertion sometimes known as Haemer's
conjecture (Brouwer and Spence 2009, Wang and Wang 2024).

# Haemers Conjecture

## See also

Cospectral Graphs, Determined by Spectrum, Graph Spectrum## Explore with Wolfram|Alpha

## References

Brouwer, A. E. and Spence, E. "Cospectral Graphs on 12 Vertices."*Elect. J. Combin.*, Vol. 16, No. 1, 2009. https://doi.org/10.37236/258.Haemers, W. H. "Are Almost All Graphs Determined by Their Spectrum?"

*Not. S. Afr. Math. Soc.*

**47**, 42-45, 2016.van Dam, E. R. and Haemers, W. H. "Which Graphs Are Determined by Their Spectrum?"

*Lin. Algebra Appl.*

**373**, 139-162, 2003.Wang, W. and Wang, W. "Haemers' Conjecture: An Algorithmic Perspective."

*Experimental Math.*, 10 Apr 2024.

## Cite this as:

Weisstein, Eric W. "Haemers Conjecture."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/HaemersConjecture.html