1 Introduction bold

Table 1

1.1 First part

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. (Doe & Smith, 2020 and Roe & Smith, 2025) Doe & Smith, 2020

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The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers.

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1.2 Subsection

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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.

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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 rye run python --version.

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    y = torch.tensor([3, 4]);
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Link to Google

The upheavals [of artificial intelligence] can escalate quickly and become scarier and even cataclysmic. Imagine how a medical robot, originally programmed to rid cancer, could conclude that the best way to obliterate cancer is to exterminate humans who are genetically prone to the disease.

2 Laplacian Operator on graphs

Let be a simple non-oriented graph with and .

This is a link to Figure 1 and this to Table 1

Doe & Smith, 2020