Editing Spectral graph theory (section)

{{Short description|Linear algebra aspects of graph theory}}
In [[mathematics]], '''spectral [[graph theory]]''' is the study of the properties of a [[Graph (discrete mathematics)|graph]] in relationship to the [[characteristic polynomial]], [[eigenvalue]]s, and [[eigenvector]]s 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 number|real]] [[symmetric matrix]] and is therefore [[Orthogonal diagonalization|orthogonally diagonalizable]]; its eigenvalues are real [[algebraic integer]]s.

While the adjacency matrix depends on the vertex labeling, its [[Spectrum of a matrix|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 graph invariant|Colin de Verdière number]].