The Study On D-magic Labeling Of Distance-regular Graph

The problem of magic labelings of graph is an active research field in the graph theory.The magic labelings originated from the Sedlácek's study on graph in the 1960s.Since then,the magic labelings theory of graphs has been enriching and developing.Let G denote a finite,undirected,simple connected graph with vertex set V(|V|?2),diameter d.Let D(?){0,1,…,d} be a set of distances in G,a bijection ?:V?{1,2,…|V|} is called a D-magic labeling of G if there exists a constant k such that?y?ND(x)?(y)=k for any vertex x?V,where ND(x)={y|(?)(x,y)=i,i?D}.The distance-regular graph is one of the important topics in graph theory.The study on D-magic labelings of distance-regular graph is of great significance to the development of not only distance-regular graph but also the D-magic graph theory.In this thesis,we study the D-magic labelings problem on two class of important distance-regular graphs:the Hamming graph H(n,q)and the Halved n-cube graph 1/2H(n,2).The main results are as follows:1.Let H(n,q)be the Hamming graph with q being a prime number at least 3.It is{1?-magic if and only if n?0(mod q),and is {0,1?-magic if and only if n?1(mod q).2.Let 1/2H(n,2)be the Halved n-cube graph with n? 2.It is ?1?-magic if and only if n=m2,where m is a positive integer at least 2,and is {0,1?-magic if and only if n=m2+2,where m is a positive integer.