Some Parameters Results Of (Generalized) Mycielskians

In a search for triangle-free graphs with arbitrarily large cheomatic,Mycielski devel-oped an interesting graph transformation:the Mycielskian ?(G)of a grpah G.For a graph G=(V,E),the Mycielskian of G is the graph ?(G)with vertex set V ? V' ?u,where V'={x':x?V} and edge set E?{xy':xy ? E} ?{y'u:y '? V'}.The vertex x' is called the twin of the vertex x and the vertex u is call the root of ?(G)Mycielskians have many interest-ing properties concerning various kinds of grpah parameters.The second part of this thesis is to study two parameters of Mycielskians and the complement of Mycielskian graph.The generalized Mycielskians(also known as cones over graphs)are the natural gener-alization of the Mycielski grpah(which were first introduced by Mycielski in 1955).Given a graph G and any integer m>0,one can transform G into a new graph ?m(G),the general-ized Mycielskian of G.Because the structure of generalized Mycielskians is more complex than Mycielskians,we should know about some properties about generalized Myceilskian by some parameters.So third part of this thesis is to study some parameters of complement the generalised Mycielskian graph.The text of paper main show independent and matching of the Mycielskian grpah and suggest diameter of complement of the Mycielskian graph.We also will give the result for diameter,domination,total domination,packing and open packing of complement of the generalised Mycielskian graph.