| Graph blow-up is a kind of graph technique which can construct a more complexgraph from a simple graph. The concept was proposed by three mathematicians whoare majored in combinatorics. After that, a mathematician used this technique tostudy enumerative problem of subgraphs. Recently, two algebraists gave a generalizeddefnition of graph blow-up and they used the new defnition to characterize comaximalgraphs.In this dissertation, the defnition of graph blow-ups and graph retract are listed,and the graph properties of these two graph techniques are studied. After that, thegraph properties of comaximal graph and co-maximal ideal graph are studied. Thenwe proved that if a ring with fnitely many maximal ideals, its comaximal graph andco-maximal ideal graph will be a blow-up graph of a certain Boolean graph. In theend, we take the advantage of comaximal graphs and co-maximal ideal graphs, someresults of ring theory are obtained.Some defnition related and some theorem which will be referred again and againis listed in Chapter1.In Chapter2, the defnition of graph blow-ups and graph retract are given, theproperties of these two techniques are studied which is the basics of the studies inChapter4.In Chapter3, we frst give the defnition of three kinds of graphs, and studiedthe graph properties of comaximal graphs and co-maximal ideal graphs, what is more,some further example are given.Chapter4shows the important applications of graph blow-up. In fact, if a ringwith fnitely many maximal ideals, its comaximal graph and co-maximal ideal graphcan be obtained by the blow-up of a certain Boolean graph. Using the properties ofgraph blow-up, we give some unifed proof of some theorems.In the end, we frst studied for a ring, with its co-maximal ideal graph beingcomplete bipartite graph, and we give a classifcation of rings with its co-maximalideal graph being planar graphs, and by virtue of comaximal graph and co-maximalideal graphs, some results on pure ring theory are obtained. |
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