The independence and independent dominating numbers of the total graph of a finite commutative ring
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Let $R$ be a finite commutative ring with nonzero unity and let $Z(R)$ be the zero divisors of $R$. The total graph of $R$ is the graph whose vertices are the elements of $R$ and two distinct vertices $x,y\in R$ are adjacent if $x+y\in Z(R)$. The total graph of a ring $R$ is denoted by $\tau (R)$. The independence number of the graph $\tau (R)$ was found in \cite{Nazzal}. In this paper, we again find the independence number of $\tau (R)$ but in a different way. Also, we find the independent dominating number of $\tau (R)$ . Finally, we examine when the graph $\tau (R)$ is well-covered.
Abstract
Let $R$ be a finite commutative ring with nonzero unity and let $Z(R)$ be the zero divisors of $R$. The total graph of $R$ is the graph whose vertices are the elements of $R$ and two distinct vertices $x,y\in R$ are adjacent if $x+y\in Z(R)$. The total graph of a ring $R$ is denoted by $\tau (R)$. The independence number of the graph $\tau (R)$ was found in \cite{Nazzal}. In this paper, we again find the independence number of $\tau (R)$ but in a different way. Also, we find the independent dominating number of $\tau (R)$ . Finally, we examine when the graph $\tau (R)$ is well-covered.
참고문헌 (15)
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