| Summary: | 碩士 === 國立中正大學 === 應用數學研究所 === 101 === Let $R$ be a commutative ring. We use $Z(R)$, $\Reg(R)$ and $\Nil(R)$ to
indicate the sets of zero-divisors, regular elements and nilpotent elements in
$R$, respectively. In this thesis, we shall introduce and
investigate the \emph{total graph} $T(\Gamma(R))$ of $R$. It is
the (undirected) graph with all elements of $R$ as vertices, and for
distinct $x, y \in R$, the vertices $x$ and $y$ are adjacent if and only if
$x+y \in Z(R)$. It is an important problem for us to discuss the
planarity of $T(\Gamma(R))$. We also study the three subgraphs $Z(\Gamma(R))$,
$\Reg(\Gamma(R))$, and $\Nil(\Gamma(R))$ of $T(\Gamma(R))$, with vertices
$Z(R)$, $\Reg(R)$, and $\Nil(R)$, respectively.
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