Paley and Peisert graphs over finite fields, and their generalizations

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dc.contributor.authorBhowmik, Anwita
dc.date.accessioned2024-01-10T11:38:39Z
dc.date.available2024-01-10T11:38:39Z
dc.date.issued2023
dc.descriptionSupervisor: Barman, Rupam
dc.description.abstractThis 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.
dc.identifier.otherROLL NO.186123003
dc.identifier.urihttps://gyan.iitg.ac.in/handle/123456789/2515
dc.language.isoen
dc.relation.ispartofseriesTH-3257
dc.subjectPaley Graphsen_US
dc.subjectPeisert Graphsen_US
dc.subjectFinite Fieldsen_US
dc.subjectDirichlet Charactersen_US
dc.subjectCharacter Sumen_US
dc.subjectCliqueen_US
dc.subjectQuadratic Residueen_US
dc.subjectHypergeometric Functions Over Finite Fieldsen_US
dc.titlePaley and Peisert graphs over finite fields, and their generalizations
dc.typeThesis
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