Near automorphisms of powers of a path

Let G be a connected graph with vertex set V ( G ) , and f a permutation on V ( G ) . Define δ f ( x , y ) = | d ( x , y ) − d ( f ( x ) , f ( y ) ) | and δ f ( G ) = ∑ δ f ( x , y ) , where the sum is taken over all unordered pairs x , y of distinct vertices of G . Denote by π ( G ) the smallest positive value of δ f ( G ) among all permutations on V ( G ) . A permutation f on V ( G ) is called a near automorphism of G if δ f ( G ) = π ( G ) . The k th power of G , written as G k , also has V ( G ) as vertex set, but x and y are adjacent in G k whenever d ( x , y ) ≤ k in G . The near automorphisms of the path P n and its square P n 2 were described in Aitken (1999) and Wong et al. (2023), respectively. In this paper, we give the value of π ( P n k ) , and determine the near automorphisms of P n k when 3 ≤ k ≤ n − 2 .