Cospectral graphs, also called isospectral graphs, are graphs that share the same graph spectrum. The smallest pair of isospectral
graphs is the graph union and star graph , illustrated above, both of which
have graph spectrum (Skiena 1990, p. 85). The first example was found
by Collatz and Sinogowitz (1957) (Biggs 1993, p. 12). Many examples are given
in Cvetkovic et al. (1998, pp. 156-161) and Rücker et al. (2002).
The smallest pair of cospectral graphs is the graph union and star
graph ,
illustrated above, both of which have graph spectrum (Skiena 1990, p. 85).

The following table summarizes some prominent named cospectral graphs.

Determining which graphs are uniquely determined by their spectra is in general a very hard problem. Only a small fraction of
graphs are known to be so determined, but it is conceivable that almost all
graphs have this property (van Dam and Haemers 2003, Haemers 2016), an assertion
sometimes known as the Haemers conjecture.

Brouwer and Spence (2009) determined the numbers of cospectral simple graphs on nodes up to , giving 0, 0, 0, 0, 2, 10, 110, 1722, 51039, 2560606, 215331676,
3106757248, ... (OEIS A006608) for , 2, .... The numbers of pairs of isospectral simple graphs
(excluding pairs that are parts of triples, etc.) are 0, 0, 0, 0, 1, 5, 52, 771,
21025, ... (OEIS A099881). Similarly, the numbers
of triples of isospectral graphs (excluding triples that are parts of quadruples,
etc.) are 0, 0, 0, 0, 0, 0, 2, 52, 2015, ... (OEIS A099882).

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 12,
1993.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.Collatz,
L. and Sinogowitz, U. "Spektren endlicher Grafen." Abh. Math. Sem. Univ.
Hamburg21, 63-77, 1957.Cvetković, D. M.; Doob,
M.; and Sachs, H. Spectra
of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.Godsil,
C. D. and McKay, B. D. "Constructing Cospectral Graphs." Aeq.
Math.25, 257-268, 1982.Haemers, W. H. "Are Almost
All Graphs Determined by Their Spectrum?" Not. S. Afr. Math. Soc.47,
42-45, 2016.Haemers, W. H. and Spence, E. "Graphs Cospectral
with Distance-Regular Graphs." Linear Multilin. Alg.39, 91-107,
1995.Rücker, C.; Rücker, G.; and Meringer, M. "Exploring
the Limits of Graph Invariant- and Spectrum-Based Discrimination of (Sub)structures."
J. Chem. Inf. Comp. Sci.42, 640-650, 2002.Skiena, S.
Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, p. 85, 1990.Sloane, N. J. A. Sequences
A006608, A099881,
and A099882in "The On-Line Encyclopedia
of Integer Sequences."van Dam, E. R. and Haemers, W. H.
"Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic
Combin.15, 189-202, 2003.