Let $G$ be a connected graph and $f$ be a mapping from $V(G)$ to $S$, where $S$ is a set of $k$ colors for some positive integer $k$. The color code of a vertex $v$ of $G$ with respect to $f$, denoted by code$_G(v|f)$, is the ordered $(k+1)$-tuple $(x_0, x_1,\ldots, x_k)$ where $x_0$ is the color assigned to $v$ and where $x_i$ is the number of vertices adjacent to $v$ of color $i$ for $1 \le i \le k$, that is, $x_i = |\{uv\in E(G) \colon f(u)=i\}|$ for $1 \le i \le k$. The mapping $f$ is a recognizable coloring if code$_G(u|f) \ne$ code$_G(v|f)$ for every two distinct vertices $u$ and $v$ of $G$. The minimum number of colors needed for a recognizable coloring of $G$ is the recognition number of $G$ denoted by rn$(G)$. Our goal in this article is to give the exact value of the recognition number of the corona product $G \circ H$ of two graphs $G$ and $H$ for the cases $H=K_n$ and $n \ge |V(G)|$, or $G=K_m$ and $H=K_n$ with $m>n$. In addition, we obtain the exact value of the recognition number of the edge corona product $G \diamond H$ of $G$ and $H$ for the case that $G$ is a non-trivial graph with minimum degree at least $2$ and $H = K_n$ where $n \ge |E(G)|$. Moreover, an algorithm for computing the recognition number of graphs is presented. As an application of our algorithm, we compute the recognition number of some fullerene graphs.