Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma_{\mathrm{Ann}}(R)$, is a graph whose vertices are all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap \mathrm{Ann}(J)\neq \{0\}$ or $J\cap \mathrm{Ann}(I)\neq \{0\}$. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with $R$. Among other results, it is proved that for a Noetherian ring $R$ if $\Gamma_{\mathrm{Ann}}(R)$ is triangle free, then $R$ is Gorenstein.

Keywords: annihilator ideal graph, diameter, Clique number

MSC numbers: 13A99, 05C75, 05C69