Some New Results Of The (Generalized) Mycielskian Graph And Its Complement Graph Of G

In search for find a class of triangle-free graph with arbitrary large chromatic numbers,Mycielski put forward an interesting graph transformation in 1955^[1],It is based on figure G through a graph transformation to get a new,We call it a Mycielskian graph,as??G?.The generalized Mycielskian graph is proposed by Lam et al^[2]and that is a natural extension of??G?,denoted by?_m?G?.In terms of definition,the generalized Mycielskian graph of figure G can be seen as a cone graph on graph G.This is the interesting structure of the generalized Mycielskian graph.More and more people are beginning to study some of the parameters of this type of graph.Do the parameters of the generalized Mycielskian graph relate to the m and the root u?It can be seen from the parameters of the existing generalized Mycielskian graphs that many of the parameters change with the changes in the m.But the influence of the root point u on the parameters of the generalized Mycielskian graph is getting weaker and weaker.The study of these parameters can help us to better describe the structure,features and properties of the Mycielskian graph.The paper is devided into four chapters,In the first chapter,we first introduce the research background and development status of Mycielskian graph,generalized Mycielskian graph and their complementary distance index degree and reformulate reciprocal index,then we use in this paper some basic concepts,terminology and symbols,and lists the main results this study.In the second chapter,we discuss the degree distance index of the Mycielskian graph of graph G,and finally give the upper bounds of the degree distance index of the graph.In the third chapter,we study the reformulated reciprocal index of the complement of?generalized?Mycielskian graph,and present the exactly formulae of the index in term of the order,size and M₁?G?of G.In the fourth chapter,we studied the Hamiltonicity and factor criticality of the generalized Mycielskian.