Spectral analysis of the affine graph over the finite ring |
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| Authors: | Jason Bell Marvin Minei |
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| Affiliation: | a Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada V5A 1S6
b Department of Mathematics, University of California, Irvine, CA 92697-3875, United States |
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| Abstract: | We apply two methods to the block diagonalization of the adjacency matrix of the Cayley graph defined on the affine group. The affine group will be defined over the finite ring Z/pnZ. The method of irreducible representations will allow us to find nontrivial eigenvalue bounds for two different graphs. One of these bounds will result in a family of Ramanujan graphs. The method of covering graphs will be used to block diagonalize the affine graphs using a Galois group of graph automorphisms. In addition, we will demonstrate the method of covering graphs on a generalized version of the graphs of Lubotzky et al. [A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988) 261-277]. |
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| Keywords: | Affine group over the finite ring Cayley graphs Eigenvalues Adjacency matrix Irreducible representations Covering graphs Ramanujan graphs |
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