Abstract
An m-L (p, p - 1, ...,1) labeling of a simple graph g is a mapping tau from the vertex set V (g) to {0, 1, ..., m} such that vertical bar tau (u) - tau(v)vertical bar >= p + 1- b (u, v), for all u, v epsilon V (g), where length of the shortest route connecting u and v is represented by b (u, v). The smallest m for which there exists a m-L (p, p - 1, ...,1) labeling of g is known as the L (p, p - 1, ...,1) labeling number of and it is described by lambda(p) (g). We define m-L' (p, p - 1, ...,1) as the same as the m-L (p, p - 1, ...,1) labeling if tau is one to one. The L' (p, p - 1, ...,1) labeling number of g represented by lambda (p, p - 1, ...,1), (g) and is called minimum span. In this paper, we prove that the circulant graphs with specified generating sets admit m-L (p, p - 1, ...,1) and m-L' (p, p - 1, ...,1) labeling and also find lambda(p) (g) and lambda (p, p - 1, ..., 1)' (g). Moreover, we find the L (p, p - 1, ...,1) labeling number of any simple graph with diameter less than p.