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- W1567270830 abstract "AbstractThe purpose of this paper is to classify nite graphs satisfyingcertain spectral bounds and give explicit methods for understandingasymptotic behavior of the spectrum. Let Xbe a nite k-regular graphand 1 (X) the second largest eigenvalue of its adjacency matrix. It fol-lows from the well-known Alon{Boppana Theorem, that for any >0there are only nitely many such Xwith 1 (X) <(2 )pk 1. Wee ectively implement Serre’s quantitative version of this result to proveseveral niteness results for the number of vertices in a k-regular graphwith 1 (X) <(2 )pk 1. For example, we give a complete classi- cation of the 3- and 4-regular graphs with 1 (X) 1. Our methodsallow one to compute explicit bounds for any kand . 1 Introduction The purpose of this paper is to classify nite graphs satisfying certain spec-tral bounds and give explicit methods for understanding asymptotic behav-ior of the spectrum. We begin with notation that will be used throughout.Fix kand let Xbe a nite connected k-regular graph with nvertices. If A" @default.
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- W1567270830 date "2014-01-01" @default.
- W1567270830 modified "2023-10-07" @default.
- W1567270830 title "Finiteness theorems in spectral graph theory" @default.
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