Abstract

Let $R$ be a commutative ring with $Nil(R)$ its ideal of nilpotent elements, $Z(R)$ its set of zero-divisors, and $Reg(R)$ its set of regular elements. In this paper, we introduce and investigate the $total\ graph$ of $R$, denoted by $T(\Gamma(R))$. 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 $x+y \in Z(R)$. We also study the three (induced) subgraphs $Nil(\Gamma(R)), Z(\Gamma(R))$, and $Reg(\Gamma(R))$ of $\Gamma(\Gamma(R))$, with vertices $Nil(R)$, $Z(R)$, and $Reg(R)$, respectively.



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