## The Laplacian spectrum of a graph.(English)Zbl 1058.05048

Summary: Let $$G = (V, E)$$ be a simple graph. Denote by $$D(G)$$ the diagonal matrix of its vertex degrees and by $$A(G)$$ its adjacency matrix. Then, the Laplacian matrix of $$G$$ is $$L(G) = D(G) - A(G)$$. The first and second section of this paper contain an introduction and some known results, respectively. The third section is devoted to properties of the Laplacian spectrum. The fourth section contains a characterization of graphs. The fifth section relates the Laplacian eigenvalues with the graph structure.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)

### Keywords:

Graph; Laplacian matrix; Largest eigenvalue; Upper bound
Full Text:

### References:

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