The diameter of power graphs of symmetric groups

The power graph P(G) of a group G is a simple graph whose vertex-set is G and two vertices x and y in G are adjacent if and only if $y = x^m$ or $x = y^m$ for some positive integer m. The proper power graph of G denoted by p*(G) is a graph which is obtained by deleting the vertex 1 (the identity element of G). It is proved that if n $\geq$ 9 and neither n nor n-1 is a prime, then P*( $S_n$) is connected. In this paper, we prove that if $n\geq 18$ and neither n nor n-1 is a prime, then d(P*( $S_n$))$\leq$10