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 SpectrumExplore 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