Spectral Graph Theory

See: Spectral Graph Theory, Spectral Graph, Affinity Matrix, Adjacency Matrix, Laplacian Matrix, Eigenvector.

References

http://en.wikipedia.org/wiki/Spectral_graph_theory
- 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. 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 {K_1,4, 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

Dragoš M. Cvetković, Michael Doob, Horst Sachs, A. Torgasev, Recent Results in the Theory of Graph Spectra (Annals of Disrete mathematics series, North-Holland) (1988) ISBN 0444703616

Collatz, L. and Sinogowitz, U. "Spektren endlicher Grafen." Abh. Math. Sem. Univ. Hamburg 21, 63–77, 1957.