Near automorphisms of cycles

Let f be a permutation of V (G). Define f (x, y)=|dG(x, y)−dG(f (x), f (y))| and f (G)= ∑ f (x, y) over all the unordered pairs {x, y} of distinct vertices of G. Let (G) denote the smallest positive value of f (G) among all the permutations f of V (G). The permutation fwith f (G)= (G) is called a near automorphism ofG. In this paper, we study the near automorphisms of cycles Cn and we prove that (Cn)= 4 n/2 − 4, moreover, we obtain the set of near automorphisms of Cn. © 2007 Elsevier B.V. All rights reserved.