Analisis Automorfisma Graf Pembagi-nol dari Ring Komutatif dengan Elemen Satuan
DOI:
https://doi.org/10.29303/emj.v1i1.11Keywords:
Ring, Graph Automorphism, star zero-divisor graph, Complete zero-divisor graph, Complete bipartite, zero-divisor graphAbstract
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.References
Anderson, D. D. dan Philip S. Livingstone, 1999, The Zero-Divisor Graph of a commutative Ring, Jurnal of Algebra, 211, Mathematic Departement, The University of Tenessee, Knoxvile.
Arifin, A., 2000, Aljabar, ITB Bandung press, Bandung.
Munir, R., 2009, Matematika Diskrit Edisi Ketiga, Informatika, Bandung.
Suryoto, 2011, Automorfisma Graph, Jurnal Matematika dan Komputer, 4 No. 3, Jurusan Matematika FMIPA UNDIP, Semarang.
Wicaksono, S. A. dan Soleha, 2013, Kajian Sifat-Sifat Graf Pembagi-nol dari Ring Komutatif dengan Elemen Satuan, Jurnal Sains dan Seni Pomits, Vol.2 No.1, Jurusan Matematika FMIPA ITS, Surabaya.
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