Research On Arithmetic Labeling Of The Circulating Ring

The graph labeling is one of the important part of the graph theory.In the graph labeling,there are kinds of grades,which main divides for two parts ,one is the additive,one is the supplementarity,example ,the graceful graph is the supplementarity.In this paper,we make a important research on the additive labelings, the arithmetic labeling ,which gains lots of reseach results when B.D.Achaya and S.M.Hegde the arithemetic labeling in 1990,whatmore,the people also pay attention to it. we reseach the arithmetic labeling of the circulating rings which composed by rings.For G = (V ,E), if there is a mapping f ( called the vertices v of the label ) from V (G ) to the set of nonnegative integer N 0, meet:(1) f (u)≠f(v),which u≠v,and u, v∈V(G);(2) { f (u)+ f(v)]uv∈E(G)} ={k , k +d,Λ, k+(q-1)d}. Then we call graph G is a ( k, d )--arithmetic graph.In this paper,we make up the circulating rings by lots of rings,and try to work out the arithmetic labeling of it ,the main job in this paper is reseach the arithmetic labeling of the circulating rings multipled by four which is with a same point,and the arithmetic labeling of the circulating rings made up by eight points with different points. I made a primaty job in this paper is:(1) we proved the arithmetic labeling of the F_{m ,4} and F_{m ,6},futher more ,we also proved the arithmetic labeling of the graph C_{8 ,i,n},in which, i =1,2,3,4.(2) We can get the arithmetic labeling of a kind of simliargraph like F_{n ,4}F_{n ,8}F_{n ,12},in last we work out the arithmetic labeling of this kind of graph F_{n , min} which m = 4k.(3) We lead in another kind of the grade ,the anti-arithmetic labeling,which move the labeling of the point to the edge,then make the labeling of the point which induced by the labeling of the edge can form the arithmetic progression.we also bing up the concept of the anti-arithmetic labeling and anti-arithmetic graph,in addition,we gains some reseach results of the arithmetic graph.