Let R be a ring with identity. The nil-clean graph of R is a graph, denoted by GNC(R), whose vertex-set is the set R, and where two distinct vertices x and y are adjacent if and only if x + y is a nil-clean element of R. An element r ∈ R is called a nil-clean element if it can be decomposed a sum of an idempotent and a nilpotent element of R. Let G be a finite undirected graph. An automorphism φ of G is a permutation on the vertex-set V(G) such that the graph preserves adjacency, that is, φ(v1) is adjacent to φ(v2) if and only if v1 is adjacent to v2. The set of all automorphisms of G together with the composition operation of permutations forms the automorphism group of G. In this paper, we firstly compute the order of the automorphism groups of nil-clean graphs for the ring ℤn. And then we determine the structure of the automorphism groups of GNC(ℤn) for n = pk, pq, 2kpl, where p, q are distinct primes and k, l are positive integers.
