Arithmetic Labeling Of Several Graphs

Graph theory is a branch of mathematics in recent years, one of the fastest in the development of a branch, and the graph labeling is an important research direction in graph theory. Graph there are many ways of labeling, in this paper,we research arithmetic labeling. Arithmetic labeling is proposed by the Indian scholar B. D. Achaya and S. M. Hegde in1990.In this paper, we research the arithmetic labeling of two new trees, flower trees and circulating rings.For G=(V, E), if there is a mapping/(called the vertices v of the label) from V(G) to the set of nonnegative integer No,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,…,k+(q-1)d}。Then we call graph G is a (k,d)-arithmetic graph.In this paper,I do the following work is:(1) We constructed two new trees, T3k and T2k, we proved the arithmetic labeling of them. We also proved the arithmetic labeling of flower trees. And we proved the three types of trees is odd arithmetic graphs.(2) We proved the arithmetic labeling of the graph C12,i,n, in which, i=1,2,3,4,5,6. Furthermore, we proved the arithmetic labeling of the graph C4k, i,n, and we proved this graph is odd arithmetic graph.