# Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory

@article{Alon1986EigenvaluesGE, title={Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory}, author={Noga Alon}, journal={Combinatorica}, year={1986}, volume={6}, pages={207-219} }

AbstractExpanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs are proved using the eigenvalues of their adjacency matrices.These graphs enable us to improve previous results on a parallel sorting problem that arises in structural modeling, by…

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## References

SHOWING 1-10 OF 60 REFERENCES

Non-existence of one-dimensional expanding graphs

- Mathematics, Computer Science22nd Annual Symposium on Foundations of Computer Science (sfcs 1981)
- 1981

This paper proves that for any finite set of onedimensional linear mappings with rational coefficients, the graph they define by their restriction to Zn is not an expanding graph, and shows that shuffle exchange graphs can not be expanding graphs.

Expanders, sorting in rounds and superconcentrators of limited depth

- Mathematics, Computer ScienceSTOC '85
- 1985

Finite geometries are used to construct explicitly highly expanding graphs with essentially the smallest possible number of edges to improve significantly previous results on a parallel sorting problem, by describing an explicit algorithm to sort elements in time units using &Ogr;(<italic>n</italic><supscrpt>) processors.

Sorting and Merging in Rounds

- Mathematics
- 1982

The need for sorting algorithms which operate in a fixed number of rounds (rather than have each new comparison depend on the outcomes of all previous comparisons) arises in structural modeling.…

Eigenvalues, Expanders and Superconcentrators (Extended Abstract)

- Mathematics, Computer ScienceFOCS
- 1984

A correspondence between the eigenvalues of the adjacency matrix of a graph and its expansion properties is shown, and results on Group Representations are combined to obtain many new examples of families of linear expanders.

il , , lsoperimetric Inequalities for Graphs , and Superconcentrators

- 1985

A general method for obtaining asymptotic isoperimetric inequalities for families of graphs is developed. Some of these inequalities have been applied to functional analysis, This method uses the…

lambda1, Isoperimetric inequalities for graphs, and superconcentrators

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. B
- 1985

Parallel Sorting with Constant Time for Comparisons

- Mathematics, Computer ScienceSIAM J. Comput.
- 1981

It is concluded that with $2n^{5/3} \log n$ parallel processors n items may be sorted with all comparisons arranged in two time intervals, and the existence of sorting algorithms achieving the bounds is proved by nonconstructive methods.

Explicit Constructions of Linear-Sized Superconcentrators

- Computer ScienceJ. Comput. Syst. Sci.
- 1981

A Geometric Construction of a Superconcentrator of Depth 2

- Computer Science, MathematicsTheor. Comput. Sci.
- 1984

An N-superconcentrator of depth 2, with 3 N32+O(N1712) edges, is constructed by essentially duplicating the lines vs. points incidence graph of a projective plane by the following theorem.