The Spectral Characterization Of The Pancyclicity Of Graphs

It has always been a difficult problem in graph theory to determine whether a graph is a pancyclic or not.Two kinds of questions have been asked concerning pancyclic graphs.First,what is the minimum number of edges,or what are the degree properties,required in order to guarantee that graph is pancyclic;in particular,how much stronger must a sufficient condition for the Hamiltonian property be to guarantee the pancyclic property?Second,what are the smallest pancyclic graphs for a given value of n?In recent decades,a large number of experts and scholars at home and abroad have given a series of sufficient conditions for pancyclic graphs,but in these sufficient conditions,the degree condition and the neighborhood union condition are basically the conditions.Therefore,there are not too many conclusions about the spectral conditions of pancyclic graphs.Firstly,this paper briefly introduces the history of graph theory,the brief background of graph theory,the basic concepts of this paper,as well as the professional terms and symbols.Then introduces the research progress of pancyclic graphs,mainly giving some degree conditions,neighborhood union conditions,edge conditions and spectral conditions of pancyclic graphs in recent decades;Finally,the main research results of this paper are introduced,and three spectral conditions of pancyclic graphs are given:1.Let G be a connected graph on n?5 vertices with ??2.If ?(G)>?n2-5n+7?,then G is pancyclic graph unless G is bipartite graph or G ?{K2? K3,K2?(K2,K2),K3?K4).32.Let G be a connected graph on n?5 vertices with ??2.If q(G)?2n-5?3/n-1,then G is pancyclic graph unless G is bipartite graph or G ?{K2? K3,K2?(K2,K2),K33.Let G be a connected graph on n>9 vertices with ??2.If ?(G)?n-3,then G is pancyclic graph unless G ?{K2?(Kn-4+K2),K1?(Kn-3+K2)}.