Some results on the maximal graph of commutative rings

Abstract

Let $\left(\mathrm{\Gamma }\right(R){)}^c$ be the maximal graph of a commutative ring with identity $R$ whose vertices are elements of $R$. Two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb\ne R$ or there exists a maximal ideal of $R$ containing both them. In this paper, we study some properties of two (induced) subgraphs $\left({\mathrm{\Gamma }}_2\right(R){)}^c$ and $\left(G\right(R){)}^c$ of $\left(\mathrm{\Gamma }\right(R){)}^c$.