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Cospectral Graphs

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CospectralGraphs

Cospectral graphs, also called isospectral graphs, are graphs that share the same graph spectrum. The smallest pair of isospectral graphs is the graph union C_4 union K_1 and star graph S_5, illustrated at left above, both of which have graph spectrum (-2)0^32 (Skiena 1990, p. 85). Regular cospectral graphs are of particular interest, and the smallest regular graphs are the two pairs of quartic graphs illustrated above center and right (Cvetković et al. 1998, p. 398; van Dam and Haemers 2003).

The first example of cospectral graphs 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 following table summarizes some prominent named cospectral graphs.

ncospectral graphs
126-antiprism graph, quartic vertex-transitive graph Qt19
16Hoffman graph, tesseract graph
16(4,4)-rook graph, Shrikhande graph
2525-Paulus graphs
2626-Paulus graphs
28Chang graphs, 8-triangular graph
70Harries graph, Harries-Wong graph

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 n nodes up to n=12, giving 0, 0, 0, 0, 2, 10, 110, 1722, 51039, 2560606, 215331676, 3106757248, ... (OEIS A006608) for n=1, 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).

See also

Determined by Spectrum, Graph Eigenvalue, Graph Spectrum, Hoffman Graph, Isospectral Manifolds, Shrikhande Graph

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References

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. Hamburg 21, 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.

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Cospectral Graphs

Cite this as:

Weisstein, Eric W. "Cospectral Graphs." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CospectralGraphs.html

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