Isospectral graphs

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 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 perfect 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 have equal multisets of eigenvalues.

Cospectral graphs need not be isomorphic, but isomorphic graphs are always cospectral.

A graph ${\displaystyle G}$ is said to be determined by its spectrum if any other graph with the same spectrum as ${\displaystyle G}$ is isomorphic to ${\displaystyle G}$.

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