Paley and Peisert graphs over finite fields, and their generalizations
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Date
2023
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Abstract
This thesis is mainly devoted to the computation of the number of cliques of certain Cayley graphs, namely the Paley- type graphs, Peisert graphs and Peisert-like graphs. Barring the case of the Peisert graphs, the focus is on the number of cliques of orders three (triangles) and four. Let q be a prime power such that q 1 (mod 4). The Paley graph of order q is the graph with vertex set as the nite eld Fq and edges de ned as, ab is an edge if and only if a b is a non-zero square in Fq. The rst part of this thesis involves de ning a generalization of the Paley graph, called the Paley-type graph on the commutative ring Zn for certain values of n, precisely n = 2sp 1 1 p k k , where s = 0 or 1, i 1, where the distinct primes pi satisfy pi 1 (mod 4) for all i = 1; : : : ; k. For such n, we de ne the graph with vertex set Zn and edges de ned as, ab is an edge if and only if a b is a square in the set of units of Zn. We look at some properties of this graph. For primes p 1 (mod 4), Evans, Pulham and Sheehan computed the number of complete subgraphs of order four in the Paley graph. Recently, Dawsey and McCarthy found the number of triangles and complete subgraphs of order four in the generalized Paley graph of prime power order. We nd the number of triangles and complete subgraphs of order four in the Paley-type graph successively for n = p (p 1 (mod 4) being a prime and 1) and for general n, using character sums and combinatorial methods.
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Supervisor: Barman, Rupam
Keywords
Paley Graphs, Peisert Graphs, Finite Fields, Dirichlet Characters, Character Sum, Clique, Quadratic Residue, Hypergeometric Functions Over Finite Fields