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A generic question:

are there any spectral techniques to estimate the genus of a graph? I am interested in complete balance multipartite graph.

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@Mohsen: A related post here on cstheory: But I guess there are much stronger properties hold on complete multipartite graphs. – Hsien-Chih Chang 張顯之 Feb 5 '11 at 15:48

Yes, there are techniques. For graphs of fixed genus and $n$ vertices, the second lowest eigenvalue of the laplacian is of order $O(1/\sqrt{n}),$ where the hidden constant depends on the genus (in an explicit way -- this follows from the Cheeger inequality and the separator theorems of Lipton and Tarjan, see eg, the paper of Spielman and Teng called "Spectral partitioning works). The dependence on the genus can be made quite explicit, so if you do that, you will get a lower bound on the genus in terms of the size of the graph and $\lambda_2.$

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I believe that this is the subject of Jon Kelner's paper Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus. In particular, he proves a lower bound of the form $O(g/n)$ for $\lambda_2$, resolving a conjecture of Spielman and Teng.

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