這篇碩士論文裡，我們首先討論有限環的零因子圖。結果顯示一個零因子圖的導出子圖是完全圖且比原來的零因子圖少一個點的話，那麼原來的零因子圖就必須是完全圖。另外確認了零因子圖一定不包含某種特殊的導出子圖。章節最後這些結果被用來確認所有包含五個點的有限交換環的零因子圖種類的可能性。
第二個部分我們研究關於有限交換環上矩陣的零因子圖。矩陣本身是非交換環，其零因子圖是有向圖。最後，對於一個點的鄰域，我們對其繪製零因子圖並得到一些有趣的結果。其圖必定連通而且直徑不大於四。

In this thesis, we first consider the zero-divisor graph of a finite commutative ring $R$. We show that if there is a complete subgraph of the zero-divisor graph Γ(R), which is obtained by deleting a vertex from Γ(R), then Γ(R) is complete. We also find that Γ(R) admits no subgraphs of certain special type. These results are used to determine all possible zero-divisor graphs of finite commutative rings with five vertices.
The second part of this thesis studies the zero-divisor graphs Γ(Mn(R)) of matrix rings over finite commutative ring, Mn(R), which is a directed graph. In particular, we found some interesting results about the closed neighbourhood of a given vertex in Γ(Mn(R)).

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