Automorphism of complete 2,3,4- ary trees

Abstract

Let G be a simple graph with vertex set V(G) and edge set E(G). Automorphism of G is an isomorphism from G to itself. In other words, an automorphism of G is a permutation φ of V(G), which has the property that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G), for every u, v ∈ V(G) that is φ preserves adjacency. The complete n-ary tree is a rooted-tree where its every internal vertex has exactly n children. The height of a rooted tree is the number of edges on longest path connecting the root to a leaf (a vertex of degree one). In this paper, we determine the number of automorphism of complete 2,3,4-ary trees with height h, for any positive integer h. We also conjecture the general formula to determine the number of automorphism of complete n-ary trees, for any n ≥ 5.