In a forthcoming paper by A. Atmaca and A. Yavuz Oruc, it has been proved that the number of unlabeled graphs with $n$ left vertices and $r$ right vertices (denoted by $|B_u(n,r)|$) satisfies the following inequality

$$\frac{{r+2^n-1\choose r}}{n!}\le |B_u(n,r)| \le 2\frac{{r+2^n-1\choose r}}{n!}, n < r. $$

Given that $|B_u(n,r)|= |B_u(r,n)|$, the following inequality holds as well when $n > r:$

$$\frac{{n+2^r-1\choose n}}{r!}\le |B_u(n,r)| \le 2\frac{{n+2^r-1\choose n}}{r!}, n > r. $$

Note that the upper bound is twice as large as the lower bound. Tightening the constant factors, i.e., 1 and 2 on the lefthand and righthand side of the inequality remains open.

The same authors also provided the following exact formulas for $|B(n,r)|$ when $n = 2$ and $n = 3.$

$$|B_u(2,r)|\!=\! \frac{2r^{3}+15r^{2} + 34r + 22.5 + 1.5\left ( -1 \right )^{r}}{24}, r=0,1,2,...$$

$\,$

$|B_{u}\left(3,r \right)|\! =
\left\{\begin{matrix}
\frac{1}{6}\left [ \binom{r+7}{7}\! + \frac{3\left ( r+4 \right )\left ( 2r^{4}+32r^{3}+172r^{2} + 352r
+ 15\left ( -1 \right )^{r} +225 \right )}{960} +\! \frac{2(r^{3}+12r^{2}+45r+54)}{54} \right ] &\!\!\!\!\!\!\text{if}\, r \bmod\!\!
\text{ } 3 = 0, \\
\\
\frac{1}{6}\left [ \binom{r+7}{7}\! + \frac{3\left ( r+4 \right )\left ( 2r^{4}+32r^{3}+172r^{2} + 352r
+ 15\left ( -1 \right )^{n} +225 \right )}{960} +\! \frac{2(r^{3}+12r^{2}+45r+50)}{54} \right ] &\!\!\!\!\!\!\!\text{ if}\, r \bmod\!\!
\text{ } 3 = 1, \\
\\
\frac{1}{6}\left [ \binom{r+7}{7}\! + \frac{3\left ( r+4 \right )\left ( 2r^{4}+32r^{3}+172r^{2} + 352r
+ 15\left ( -1 \right )^{r} +225 \right )}{960} +\! \frac{2(r^{3}+12r^{2}+39r+28)}{54} \right ] &\!\!\!\!\!\!\!\!\! \text{if}\, r \bmod\!\!
\text{ } 3 = 2. \!\!\!\! \end{matrix}\right.
$

Ref: Abdullah Atmaca and A. Yavuz Oruc. "On The Size Of Two Families Of Unlabeled Bipartite Graphs." AKCE International Journal of Graphs and Combinatorics. https://doi.org/10.1016/j.akcej.2017.11.008.