Study on The Extremal Graphs, Hypergraphs and Symmetric Structures with applications

Abstract

The thesis exposes concept of the extremal graph, hypergraph and symmetric structure with applications. We construct the symmetric structure for extremal graph like: bipartite graph, graph excluding cycles and hypergraph like: hypergraph with symmetric, finite projective plane which represents the hypergraph. The thesis proved the Erdos-Stone theorem and give the extension of Erdos-Stone-Bollobas theorem and also the result of Erdos-Stone theorem can be viewed as a considerable extension of Turan's theorem. The study aims to give symmetric structure of extremal graph, hypergraph and relationship between them. Through regular bipartite multigraphs of degree m, skew cubic bipartite multigraphs, isomorphism problem for hypergraph, extremal problem of graph and hypergraph reconstruction. The study shows a relation between colouring graph and four colour problem and hypergraph colouring, and also chromatic polynomials with bipartite graph and k-partite graph. The study gives a deep concepts and nation of colouring problem and its development. The study gives some applications from Latina squares and applications of general theory of hypergraph like: hypergraph theory and system modeling for engineering, hypergraph covering and packing, a game theoretic approach and generalized hypergraph incidence game. Which describes the relationship of these applications with other sciences.