## Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus (2004)

Venue: | in Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, ACM |

Citations: | 15 - 3 self |

### BibTeX

@INPROCEEDINGS{Kelner04spectralpartitioning,,

author = {Jonathan A. Kelner},

title = {Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus},

booktitle = {in Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, ACM},

year = {2004},

pages = {455--464},

publisher = {Press}

}

### OpenURL

### Abstract

Abstract. In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [13] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O ( p g/n) and vertex bisectors of size O ( √ gn) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces. 1. Introduction. Spectral

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Citation Context ...aper, we shall concern ourselves exclusively with λ2, also known as the algebraic connectivity or Fiedler value of G. We call the vector v2 1 Djidjev claimed in a brief note to have such an algorithm =-=[9]-=-, but it has never appeared in the literature. 2sthe Fiedler vector of G. As we shall see in Section 2.2, the Fiedler value of a graph is closely related to how well connected the graph is. A differen... |

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Citation Context ...ric, it is guaranteed to have an orthonormal basis of real eigenvectors and exclusively real eigenvalues. Let λ1 ≤ λ2 ≤···≤λn be the eigenval1Djidjev claimed in a brief note to have such an algorithm =-=[10]-=-, but it has never appeared in the literature.884 JONATHAN A. KELNER ues of L(G), and let v1,...,vn be a corresponding orthonormal basis of eigenvectors. For any G, the all-ones vector will be an eig... |