Let $R$ be a commutative ring with $1\neq 0$ and the additive group $R^+$. The several graphs on $R$ have been introduced by many authors, among zero-divisor graph $\Gamma_1(R)$, co-maximal graph $\Gamma_2(R)$, annihilator graph $\AG(R)$, total graph $\T(\Gamma(R))$, cozero-divisors graph $\Gamma_c(R)$, equivalence classes graph $\Gamma_{E}(R)$ and the Cayley graph $\Cay(R^+ ,\Z^*(R))$. Shekarriz et al. (J. Communications in Algebra, 40 (2012) 2798-2807) given some condition under which total graph is isomorphic to $\Cay(R^+ ,\Z^*(R))$. Badawi (J. Communications in Algebra, 42 (2014) 108-121) shows that when $R$ is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if $R$ has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring $R$ with $|\Max (R)|=n \geq 3$, $ \Gamma_1(R) \simeq \Gamma_2(R)$ if and only if $R\simeq \mathbb{Z}^n_2$; if and only if $\Gamma_1(R) \simeq \Gamma_{E}(R)$. Also the annihilator graph is identical to the cozero-divisor graph if and only if $R$ is a Frobenius ring.
