On Š℘,℘−1,…,1 Labelings of Circulant Graphs

An m-Š℘,℘−1,…,1 labeling of a simple graph G is a mapping τ from the vertex set ∨G to 0,1,…,m such that τu−τv≥℘+1−du,v, ∀u,v∈∨G, where length of the shortest route connecting u and v is represented by du,v. The smallest m for which there exists a m-Š℘,℘−1,…,1 labeling of G is known as the Š℘,℘−1,…,1 labeling number of G, and it is described by λ℘G. We define m-Š′℘,℘−1,…,1 as the same as the m-Š℘,℘−1,…,1 labeling if τ is one to one. The Š′℘,℘−1,…,1 labeling number of G represented by λ℘,℘−1,…,1′G and is called minimum span. In this paper, we prove that the circulant graphs with specified generating sets admit m-Š℘,℘−1,…,1 and m-Š′℘,℘−1,…,1 labeling and also find λ℘G and λ℘,℘−1,…,1′G. Moreover, we find the Š℘,℘−1,…,1 labeling number of any simple graph with diameter less than ℘.