Analisis Automorfisma Graf Pembagi-nol dari Ring Komutatif dengan Elemen Satuan

Abstract

Zero-divisor graphs of a commutative ring with identity has 3 specific simple forms, namely star zero-divisor graph, complete zero-divisor graph and complete bipartite zero-divisor graph. Graph automorphism is one of the interesting concepts in graph theory. Automorphism of  graph G is an isomorphism from graph G to itself. In other words, an automorphism of a graph G is a permutation φ of  the set points V(G) which has the property that (x,y) in E(G)  if and only if (φ(x),φ(y)) in E(G), i.e. φ preserves adjacency.This study aims to analyze the form of zero-divisor graph automorphisms of a commutative ring with identity formed. The method used in this study was taking sampel of each zero-divisor graph to represent each graph. Thus, pattern and shape of automorphism of each graph can be determined. Based on the results of this study, a star zero-divisor graph with pattern K_1,(p-1), where p is prime, has (p-1)! automorphisms, a complete zero-divisor graph with pattern K_(p-1), where p is prime, has (p-1)!  automorphisms, and a complete bipartite zero-divisor graph with pattern K_(p-1),(q-1), where p is prime, has (p-1)!(q-1)! automorphisms, when p not equals to q  and 2((p-1)!(q-1)!) automorphisms  when p=q.