Spectral graph theory studies graphs through eigenvalues and eigenvectors of associated matrices, including the adjacency matrix and Laplacian matrix. A graph eigenvalue usually refers to the adjacency-matrix spectrum unless another matrix is specified.
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Spectral graph theory is a branch of mathematics that studies the properties of graphs through the eigenvalues and eigenvectors of matrices associated with them. These matrices include the adjacency matrix, the degree matrix, and the Laplacian matrix, among others. Spectral graph theory connects combinatorial properties of graphs with linear algebra and provides powerful tools for analyzing graphs in various contexts.