## Numerical Investigation of the Spectrum for Certain Families of Cayley Graphs (1993)

Venue: | in DIMACS Series in Disc. Math. and Theor. Comp. Sci |

Citations: | 8 - 5 self |

### BibTeX

@INPROCEEDINGS{Lafferty93numericalinvestigation,

author = {John Lafferty and Daniel Rockmore},

title = {Numerical Investigation of the Spectrum for Certain Families of Cayley Graphs},

booktitle = {in DIMACS Series in Disc. Math. and Theor. Comp. Sci},

year = {1993},

pages = {63--73},

publisher = {American Mathematical Society}

}

### OpenURL

### Abstract

In this paper we extend some earlier computations [8]. In particular, the expanding behavior of Cayley graphs of PSL2 (F107) is compared with that of the Cayley graphs for the group A10 . These computations support the (up to now) unvoiced conjecture of Lubotzky that the symmetric groups and projective linear groups have asymptotically different average expanding behavior. We also give a thorough spectral analysis for a natural family of Cayley graphs which does not admit analysis by Selberg's theorem. 1 Introduction Spectral analysis and operator theory have provided some of the main tools for the recent advances in constructions of expander graphs. In particular, by exploiting the various relationships between the second largest eigenvalue of the Laplacian and the expansion coefficient of graphs, families of expanders have been constructed and analyzed. When the graphs of interest are Cayley graphs, techniques from Fourier analysis are especially useful in this analysis. In this pap...

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Citation Context ... c ffi S (ae i ): (1) For numerical analysis of the spectrum, (2.1) provides a great advantage. Standard techniques for computing eigenvalues of a square matrix of degree d require O(d 3 ) operations =-=[13]-=-. As the degree of the largest irreducible representation is bounded by j G j 1=2 , (2.1) brings the computation from O(j G j 3 ) operations to a more manageable O \Gamma j G j \Delta(max i deg(ae i )... |

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Citation Context ...X. The second-largest eigenvalue of X, denoteds1 (X), is defined to bes1 = max fi: j �� i j6=kg j �� i j : Various connectivity properties of a graph can be judged by studying its spectrum (se=-=e e.g., [1]-=-). Bounds for the diameter, expansion coefficient, and chromatic number all can be given in terms ofs1 . For details and references we refer the reader to other papers in this volume as well as the bo... |

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Citation Context ...i )) \Delta operations. In the case in which S S S \Gamma1 is a union of conjugacy of classes of G, c ffi S can be completely diagonalized and the diagonal elements computed as certain character sums =-=[3]-=-. However, if S S S \Gamma1 is not a union of conjugacy classes (which is the situation in the computations considered here), then the actual matrix representations are required to compute the spectru... |

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Citation Context ...(F p ), PSL 2 (F p ) and A n . In this section we give a very quick introduction and explanation of the representation theory of SL 2 (F p ), PSL 2 (F p ) and An . In-depth treatments can be found in =-=[4]-=- for An and [10] for SL 2 (F p ) and PSL 2 (F p ). 3.1 Representation theory for the alternating group The representation theory of the alternating groups is most easily explained by using the represe... |

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Citation Context ...for the diameter, expansion coefficient, and chromatic number all can be given in terms ofs1 . For details and references we refer the reader to other papers in this volume as well as the books ([9], =-=[11]-=-). Fourier analysis provides an extremely useful tool for the study of the spectra of Cayley graphs. Recall that if f is any complex-valued function defined on G and ae is any matrix representation of... |

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Citation Context ...F p ) and A n . In this section we give a very quick introduction and explanation of the representation theory of SL 2 (F p ), PSL 2 (F p ) and An . In-depth treatments can be found in [4] for An and =-=[10]-=- for SL 2 (F p ) and PSL 2 (F p ). 3.1 Representation theory for the alternating group The representation theory of the alternating groups is most easily explained by using the representation theory o... |

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Citation Context ...ers have been constructed and analyzed. When the graphs of interest are Cayley graphs, techniques from Fourier analysis are especially useful in this analysis. In this paper we extend the analysis of =-=[8]-=- by presenting two different computations. The first computation gives an analysis of the spectrum of the Cayley graphs of SL 2 (F p ) on the generating sets ni 1 2 0 1 j ; i 1 0 2 1 jo and ni 1 3 0 1... |

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Citation Context ...r of elements of SL 2 (F p ) is chosen at random, then with high probability (approaching 1 as p goes to infinity, at a rate of roughly 1 \Gamma O(log 2 p=p)) these elements will generate SL 2 (F p ) =-=[6]-=-. In [8] this led us to consider the following question: if two elements a; b are chosen uniformly from SL 2 (F p ) subject to the constraint that ha; bi = SL 2 (F p ), then what can be said abouts1 (... |

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Citation Context ...e asymptotics of numerical spectral analysis is less well-developed. A serious computational consideration for the symmetric groups is that the degrees of the irreducible representations grow quickly =-=[7]-=-. Thus, the determination of the eigenvalues soon dominate the computation. The symmetric group S 10 is still computationally tractable from this point of view (with largest irreducible representation... |

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Citation Context ...ere are only 82 such classes for the involution (12)(34)(56)(78). Finally, it has been observed that the action of SL 2 (F p ) on the projective line approximates the action of SL 2 (F p ) on itself (=-=[2], [8-=-]). That is, SL 2 (F p ) acts naturally on P 1 (F p ). Let ha; bi = SL 2 (F p ) and consider the graph with vertex set P 1 (F p ) and v �� w iff sv = w for some s 2 fa; bg[fa; bg \Gamma1 . The con... |

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Citation Context ...the orbit structure of the action of the generators (lifted to SL 2 (F p )) on the projective line, where the action should be transitive. This is explained fully in [8]. For G = An , we use the fact =-=[5]-=- that if a permutation group on n ? 5 points is transitive on 4-sets (4-homogeneous), then outside of a few exceptions, the group is either An or Sn . The orbit structure of the action of the group ge... |

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Citation Context ...s of PSL 2 (F p ). As for the symmetric group, the irreducible representations for SL 2 (F p ) may be effciently constructed. Explicit contructions are given in [8], using the discussions in [10] and =-=[12]-=-. In short, all representations of the principal series can be constructed as induced 1-dimensional representations from B and as such are essentially in 1-1 correspondence with the characters of the ... |