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I wonder is there any geometric interpretation of the eigenvalues ​​of the discrete Laplace operator on graphs? Maybe there is a relationship between the eigenvalues ​​and combinatorial properties of the graph?

I would like to know what information eigenvalues contain.

Thanks for the answer.

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The magic words are "spectral graph theory". Google it.

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I strongly recommend the following book as for reference:

Chung, Fan R. K. Spectral graph theory. CBMS Regional Conference Series in Mathematics, 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997.

I was learning basic notion and analysis on graphs as well. This book contains almost every topics related to analysis.

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  • $\begingroup$ Problem is, Chung's book is solely devoted to the so-called normalized Laplacian, whose spectral properties have very little in common with those of the discrete Laplacian. Mohar's survey are probably a better start. $\endgroup$
    – Delio Mugnolo
    Commented Feb 10, 2016 at 13:46

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