Spectral graph theory

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In mathematics, spectral graph theory is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated to the graph, such as its adjacency matrix or Laplacian matrix.

An undirected graph has a symmetric adjacency matrix and therefore has real eigenvalues (the multiset of which is called the graph's spectrum) and a complete set of orthonormal eigenvectors.

While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant.

Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number.

Isospectral graphs[edit]

Two graphs are called isospectral or cospectral if the adjacency matrices of the graphs have equal multisets of eigenvalues.

Isospectral graphs need not be isomorphic, but isomorphic graphs are always isospectral. The smallest pair of nonisomorphic cospectral undirected graphs is {K1,4, C4 U K1}, comprising the 5-vertex star and the graph union of the 4-vertex cycle and the single-vertex graph, as reported by Collatz and Sinogowitz[1][2] in 1957. The smallest pair of nonisomorphic cospectral polyhedral graphs are enneahedra with eight vertices each.[3]

Almost all trees are cospectral, i.e., the share of cospectral trees on n vertices tends to 1 as n grows.[4]

Isospectral graphs are also constructed by means of the Sunada method.[5]

Historical outline[edit]

Spectral graph theory emerged in the 1950s and 1960s. Besides graph theoretic research on the relationship between structural and spectral properties of graphs, another major source was research in quantum chemistry, but the connections between these two lines of work were not discovered until much later.[6] The 1980 monograph Spectra of Graphs[7] by Cvetković, Doob, and Sachs summarised nearly all research to date in the area. In 1988 it was updated by the survey Recent Results in the Theory of Graph Spectra.[8] The 3rd edition of Spectra of Graphs (1995) contains a summary of the further recent contributions to the subject.[6]

See also[edit]


  1. ^ Collatz, L. and Sinogowitz, U. "Spektren endlicher Grafen." Abh. Math. Sem. Univ. Hamburg 21, 63–77, 1957.
  2. ^ Weisstein, Eric W., "Cospectral Graphs", MathWorld.
  3. ^ Hosoya, Haruo; Nagashima, Umpei; Hyugaji, Sachiko (1994), "Topological twin graphs. Smallest pair of isospectral polyhedral graphs with eight vertices", Journal of Chemical Information and Modeling 34 (2): 428–431, doi:10.1021/ci00018a033 .
  4. ^ Schwenk, A. J. "Almost All Trees are Cospectral" In: New Directions in the Theory of Graphs (F. Harary, Ed.), Academic Press, New York, 1973, pp. 275–307.
  5. ^ Sunada, Toshikazu (1985), "Riemannian coverings and isospectral manifolds", Ann. of Math. 21: 169–186 .
  6. ^ a b Eigenspaces of Graphs, by Dragoš Cvetković, Peter Rowlinson, Slobodan Simić (1997) ISBN 0-521-57352-1
  7. ^ Dragoš M. Cvetković, Michael Doob, Horst Sachs, Spectra of Graphs (1980)
  8. ^ Cvetković, Dragoš M.; Doob, Michael; Sachs, Horst; Torgasev, A. (1988). Recent Results in the Theory of Graph Spectra. Annals of Discrete mathematics (36). ISBN 0-444-70361-6. 

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