The Exponent Set of the kth Power Graphs of Connected Simple Graphs

Let G be a connected simple graph of order n and k an integer with k ≥2. The k th power graph of G , denoted by G k , is defined as a graph with V(G k)=V(G) and for any u,v∈V(G k) (u≠v), (u,v)∈E(G k) if and only if d G(u,v)≤k . Then G k is primitive for any k≥2 . Let E(k,n)={γ(G k)|G is a connected simple graph of order n }. In this paper, we obtainE(k,n)=dkk+1≤d≤n-1, if 2≤k≤n-2, {2}, if k≥n-1.