The Existence Of 2-factor And Some A_α-spectral Property Of Graphs

Graph theory has received a rapid development and has been widely used in information theory,cybernetics,network theory,game theory,operational research and other fields in recent years.This paper considers two important fields in graph theory.The first one is the coloring problems of the graph.Graph coloring is an essential research field in graph theory,which mainly contains vertex coloring,edge coloring,face coloring,weakly edge-face coloring,where edge coloring is a classical topic among them.This paper mainly focus on the existence of the 2-factor of edge-coloring graph.We showed that every edge-colored graph G with |G|=n and |CN(u)∪CN(v)|≥4n/3+8 contains a properly colored 2-factor.The second part of this paper is about spectral theory.As an important part of graph theory and combinatorial matrix theory,spectral theory is widely used in computer science,quantum chemistry,communication network and other fields.We consider the upper bounds on the Aα-spectral radius of graph G and showed that if k is the chromatic number of graph G and ρα(G)=λ1(G)≥λ2(G)≥…≥λn(G),then ρα(G)≤2αe-(λn(G)+…+λn-k+2(G)).Also,we give some upper bounds for the second largest eigenvalue of Aα-matrix.