Applications of Spectral Graph Theory to Some Classical Problems in Combinatorics and Number Theory
Author | : Yesim Demiroğlu Karabulut |
Publisher | : |
Total Pages | : 96 |
Release | : 2018 |
ISBN-10 | : OCLC:1192471247 |
ISBN-13 | : |
Rating | : 4/5 ( Downloads) |
Download or read book Applications of Spectral Graph Theory to Some Classical Problems in Combinatorics and Number Theory written by Yesim Demiroğlu Karabulut and published by . This book was released on 2018 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: "In the first half of this thesis, we obtain sharp results for Waring's problem over general finite rings, by using a combination of Artin-Wedderburn theory and Hensel's lemma and building on new proofs of analogous results over finite fields that are achieved using spectral graph theory. We also prove an analogue of Sárközy's theorem for finite fields. In the second half of the thesis, we investigate the unit-graphs and the special unit-digraphs on matrix rings and we show that every n x n nonzero matrix over Fq can be written as a sum of two SLn-matrices when n > 1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties. We prove that if X is a subset of Mat2(Fq) with size [equation would not render] then X contains at least two distinct matrices whose difference has determinant for any [equation would not render]. Using this result we also prove a sum-product type result: if A,B,C;D[subset]Fq satisfy [equation would not render] as q[rightarrow][infinity], then (A-B)(C-D) equals all of F*q. In particular, if A is a subset of Fq with cardinality |A| > 3/2 q 3/4, then the subset (A - A)(A - A) equals all of Fq. We also recover some classical results, e.g. every element in any finite ring of odd order can be written as the sum of two units, and we also derive some character sum identities."--Page vii.