Graceful Labeling of Prime Index Graph of Group Zp × Zpn

The prime index graph π(G) of a finite group G is special type undirected simple whose vertex set subgroups G, in which two distinct vertices are adjacent if one has the other. Let p and q be primes. In this paper, we establish that cyclic p-group Zpn, abelian Zpn × Zq Zp always have graceful labeling without any condition on n using concept path or p-layer ladder size + 1.