Graceful Labeling of Prime Index Graph of Group Zp × Zpn

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