In the literature, there are many paper on assigning a graph to a ring, a group, semigroup or a module. Here, we consider some properties of graphs assigned to commutative rings.\\ The zero divisor graph of a commutative ring, which is represented by $\Gamma (R)$, is a graph with the vertices set $Z(R)\setminus\{0\}$ in which two vertices $x$ and $y$ are adjacent if $xy=0$. In this thesis, commutative rings for which $\Gamma (R)$ is planar, is considered and all finite rings for which this property is hold, are characterized. Also, all planar zero divisor graphs of infinite commutative rings are characterized.\\ Let $R$ be a commutative ring with $\ {A}(R)$ its set of ideals with nonzero annihilator. We introduce and investigate {\it annihilating ideal graph} of $R$, denoted by $\ {AG}(R)$. It is an undirected graph with vertices $\ {A}^*(R):=\ {A}(R)\setminus\{(0)\}$, in which two vertices $I$ and $J$ are adjacent if $IJ=(0)$. First, we study some finiteness conditions of ${\{AG}}(R)$. Next, we study the connectivity of ${\{AG}}(R)$. (Also, we determine the relationship between the diameter of ${\{AG}}(R)$ and $\Gamma(R)$finally,we study the coloring of the annihilating-ideal graph of rings. For a graph $G$, the chromatic number and the clique of $G$ is denoted by $\chi (G)$ and $cl(G)$, respectively. It is shown that for a reduced ring $R$ the following conditions are equivalent: (1) $\chi({\{AG}}(R)) \infty$, (2) $cl({\{AG}}(R)) \infty$, (3) $\{AG}(R)$ does not have an infinite clique and (4) $R$ has finite number of minimal primes. Moreover, if $R$ a non-domain reduced ring, then $\chi({\{AG}}(R)$ is the number of minimal primes of $R$.