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Concept
Spectral graph theory
Summary
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.
The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers.
While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one.
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.
Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if the adjacency matrices have equal multisets of eigenvalues.
Cospectral graphs need not be isomorphic, but isomorphic graphs are always cospectral.
A graph is said to be determined by its spectrum if any other graph with the same spectrum as is isomorphic to .
Some first examples of families of graphs that are determined by their spectrum include:
The complete graphs.
The finite starlike trees.
A pair of graphs are said to be cospectral mates if they have the same spectrum, but are non-isomorphic.
The smallest pair of cospectral mates is {K1,4, C4 ∪ 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 in 1957.
The smallest pair of polyhedral cospectral mates are enneahedra with eight vertices each.
Almost all trees are cospectral, i.e., as the number of vertices grows, the fraction of trees for which there exists a cospectral tree goes to 1.
A pair of regular graphs are cospectral if and only if their complements are cospectral.
A pair of distance-regular graphs are cospectral if and only if they have the same intersection array.
Cospectral graphs can also be constructed by means of the Sunada method.
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