  For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic
if the edges of $M$ are colored with the same color. The graph $G$ is called
monochromatic disconnected if any two distinct vertices of $G$ are separated by
a monochromatic edge-cut. For a connected graph $G$, the monochromatic
disconnection number (or $MD$-number for short) of $G$, denoted by $md(G)$, is
the maximum number of colors that are allowed in order to make $G$
monochromatic disconnected. For graphs with diameter one, they are complete
graphs and so their $MD$-numbers are $1$. For graphs with diameter at least 3,
we can construct $2$-connected graphs such that their $MD$-numbers can be
arbitrarily large; whereas for graphs $G$ with diameter two, we show that if
$G$ is a $2$-connected graph then $md(G)\leq 2$, and if $G$ has a cut-vertex
then $md(G)$ is equal to the number of blocks of $G$. So, we will focus on
studying $2$-connected graphs with diameter two, and give two upper bounds of
their $MD$-numbers depending on their connectivity and independent numbers,
respectively. We also characterize the
$\left\lfloor\frac{n}{2}\right\rfloor$-connected graphs (with large
connectivity) whose $MD$-numbers are $2$ and the $2$-connected graphs (with
small connectivity) whose $MD$-numbers archive the upper bound
$\left\lfloor\frac{n}{2}\right\rfloor.$ For graphs with connectivity less than
$\frac n 2$, we show that if the connectivity of a graph is in linear with its
order $n$, then its $MD$-number is upper bounded by a constant, and this
suggests us to leave a conjecture that for a $k$-connected graph $G$,
$md(G)\leq \left\lfloor\frac{n}{k}\right\rfloor$.
