Some graphs with small second eigenvalue

Abstract

For any prime,p, we construct a Cayley graph on the group,G, of affine linear transformations ofℤ/pℤ of degree 2(p−1) and second eigenvalue\(2\sqrt p \) with the following special property: the adjacency matrix of the graph is supported on the “blocks” associated to the trivial representation and the irreducible representation of sizep−1. SinceG is of orderp(p−1), the correspondingt-uniform Cayley hypergraph has essentially optimal second eigenvalue for this degree and size of the graph (see [2] for definitions). En route we give, for any integerk>1, a simple Cayley graph onp k nodes of degreep of second eigenvalue\( \leqslant (k - 1)\sqrt p \).

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The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and the Office of Naval Research under Grant N00014-87-K-0467.

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Friedman, J. Some graphs with small second eigenvalue. Combinatorica 15, 31–42 (1995). https://doi.org/10.1007/BF01294458

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Mathematics Subject Classification (1991)

  • 05 C 50
  • 68 R 10
  • 05 C 65