## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, May 08, 2014, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: May Szedlák

## Discrete Higher Dimensional Cheeger Inequalities

For graphs there exists a strong connection between spectral expansion and edge expansion. This is expressed, e.g., by the Cheeger inequality, which states that \lambda(G) \leq h(G), where \lambda(G) is the second smallest eigenvalue of the Laplacian of G and h(G) the Cheeger constant measuring the edge expansion of G. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension. Whereas for simplicial complexes, higher dimensional Laplacians were introduced already in 1945 by Eckmann, the generalization of edge expansion is not straightforward. Recently, a topologically motivated notion analogous to edge expansion was introduced by Gromov and independently by Linial, Meshulam and Wallach and by Newman and Rabinovich. It is known that for this generalization there is no higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, denoted by h(X), was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed \lambda(X) \leq h(X), where \lambda(X) is the smallest non-trivial eigenvalue of the Laplacian, for the case of k-dimensional simplicial complexes X with complete (k-1)-skeleton. Whether this inequality also holds for k-dimensional complexes with non-complete (k-1)-skeleton has been an open question. We give two different strengthenings of the inequality for arbitrary complexes.

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