  A tree $T$, in an edge-colored graph $G$, is called {\em a rainbow tree} if
no two edges of $T$ are assigned the same color. A {\em $k$-rainbow coloring}of
$G$ is an edge coloring of $G$ having the property that for every set $S$ of
$k$ vertices of $G$, there exists a rainbow tree $T$ in $G$ such that
$S\subseteq V(T)$. The minimum number of colors needed in a $k$-rainbow
coloring of $G$ is the {\em $k$-rainbow index of $G$}, denoted by $rx_k(G)$.
Graph operations, both binary and unary, are an interesting subject, which can
be used to understand structures of graphs. In this paper, we will study the
$3$-rainbow index with respect to three important graph product operations
(namely cartesian product, strong product, lexicographic product) and other
graph operations. In this direction, we firstly show if $G^*=G_1\Box
G_2\cdots\Box G_k$ ($k\geq 2$), where each $G_i$ is connected, then
$rx_3(G^*)\leq \sum_{i=1}^{k} rx_3(G_i)$. Moreover, we also present a condition
and show the above equality holds if every graph $G_i (1\leq i\leq k)$ meets
the condition. As a corollary, we obtain an upper bound for the 3-rainbow index
of strong product. Secondly, we discuss the 3-rainbow index of the
lexicographic graph $G[H]$ for connected graphs $G$ and $H$. The proofs are
constructive and hence yield the sharp bound. Finally, we consider the
relationship between the 3-rainbow index of original graphs and other simple
graph operations : the join of $G$ and $H$, split a vertex of a graph and
subdivide an edge.
