Graph Spectral Properties, And (g, F) - Factor

This paper consists of two parts. The first part is mainly about some properties of spectrum of graph.The second part is mainly about some results on fractional (g, f) -factor.About the properties of spectrum of graph, firstly we study the spectrum of an important graph-regular graph G We find out the relation among L(G) and A(G) and B(G). Then we study the spectrum of regular graph G and it's linear graph LG and total graph TG, we get the largest and the smallest eigenvalue and the scope of the middle eigenvalue. Then we apply the results getting from the studies of regular graph G to study another graph: Parsons graph Tb(d,q) in special linear group SLd(q). We find out the relation among L(Tb) and A(Tb) and B(Tb). Then we study the spectrum of Parsons graph Tb (d,q) and it's linear graph LTband get the largest and the smallest eigenvalue and the scope of the middle eigenvalue.About some results on fractional (g, f) - factor, Firstly we willintroduce the developmentof the factors of graph and give some knowledge about them. Then, on the basis of some properties and some concepts of the fractional (g,f)-factor put forward recently, we extend some concepts and bring out some new concepts specially.. And then we will define fractional (g,f,H(k))- covered graph and fractional (g,f,H(k))- strong covered graph and fractional(g,f,k)-covered graph and fractional (g,f,H(k))-deleted graph and fractional (g,f,k)-deleted graph and so on. Then we find out some important properties on fractional covered graph , fractional deleted graph and fractional uniformed graph . We get and prove thesufficient and necessary conditions on fractional (g,/)- strong covered graph, and on fractional (g,f)-strong uniformed graph. And we also study some relations between isolated toughness I(G) and fractional (g,/) - factor.