Properties Of Zero-divisor Graph Of Zn[i]

Ring theory and graph theory are two very important mathematical branches, whichare not only of great theoretical interest in themselves but also found importantapplications in many other branches of math (such as combinatorial mathematics,geometry , automata theory and coding theory, etc.). The zero divisor graph of rings,using properties of graphs to study algebraic structures, has become an excitingresearch topic in the last twenty years, leading to many fascinating results andquestions. In the past ten years, it has become a hot research field.Z_n[i] is very important ring in ring theory, often as scattered examples in abstractalgebra. By synthesizes methods of commutative algebra, abstract algebra andgraph theory, we will study some properties of zero-divisors graph of Z_n[i] and itscomplement graph.In Chapter 1 of this paper, we summarize the history of the zero-divisor graph,the background and main results of this paper. At the same time, we give thenotations and basic results of ring theory and graph theory.In Chapter 2, we determine the genus ofΓ(Z_n[i]). We give whenΓ(Z_n[i]) isperfect and in the case the chromatic number ofΓ(Z_n[i]) is determined.In Chapter 3, we completely evaluate the domination number ofΓ(Z_n[i]).Wecompute the independence number ofΓ(Z_n[i]) for some cases of n.In Chapter 4, connectivity and the genus ofΓ(Z_n[i]) are studied.