| Graph Theory is a branch of mathematics. Graph theory is also intimately related to many branches of mathematics, including group theory, matrix theory, numerical analysis, probability, topology, and combinatorics. The fact is that graph theory serves as a mathematical model for any system involving a binary relation. Partly because of their diagrammatic representation, graphs have an intuitive and aesthetic appeal. Many problems in graph theory are usually quite easy to state and explain, even for the layman, but they are too difficult to solve even for the most sophisticated mathematician.With the development of computer science and mathematics. Application of graph theory to some areas of physics, chemistry, communication science, computer technology, electrical and civil engineering, architecture, operational research, genetics, psychology, sociology, economics, anthropology, and linguistics, etc. graph theory turn into an important tool to research technology and nature science, even for the social science.Extremal theory is an important part of Graph Theory.The prime example of an extremal problem is the following: given aclass of graphs ={G1, G2,...,Gm}, determine ex(n; V), the maximum number of edges in a graph of order n not containing V as a subgraph, determine EX(n; w}, the set of graphs with order n and maximum size and not containing as a subgraph.For the extremal problems not containing polygon, for V={C4},Clapham investigated the values of ex(n)(n<21) (Journal of Graph Theory, Vol.13, no.l, (1989), 29-47), Yuansheng investigated the values of ex(n; V)(22 < n < 31 )(UTILITAS MATHEMATICA, 41, (1992), 204-210); for V={ C3,C4},Garnick investigated the values of ex(n; V) (n<24 )(Journal of Graph Theory, Vol.17, no.5, (1993), 633-645).This paper investigates the values of ex(n; V) for ={C3,C4,C5}(n<42) gives all graphs in EX(n;{C3,C4,Cs})(n<42), and shows upper bounds for all ?42. |
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