  A path in an edge-colored graph, where adjacent edges may be colored the
same, is a rainbow path if no two edges of it are colored the same. For any two
vertices $u$ and $v$ of $G$, a rainbow $u-v$ geodesic in $G$ is a rainbow $u-v$
path of length $d(u,v)$, where $d(u,v)$ is the distance between $u$ and $v$.
The graph $G$ is strongly rainbow connected if there exists a rainbow $u-v$
geodesic for any two vertices $u$ and $v$ in $G$. The strong rainbow connection
number of $G$, denoted $src(G)$, is the minimum number of colors that are
needed in order to make $G$ strong rainbow connected. In this paper, we first
investigate the graphs with large strong rainbow connection numbers. Chartrand
et al. obtained that $G$ is a tree if and only if $src(G)=m$, we will show that
$src(G)\neq m-1$, so $G$ is not a tree if and only if $src(G)\leq m-2$, where
$m$ is the number of edge of $G$. Furthermore, we characterize the graphs $G$
with $src(G)=m-2$. We next give a sharp upper bound for $src(G)$ according to
the number of edge-disjoint triangles in graph $G$, and give a necessary and
sufficient condition for the equality.
