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Solution

Let the states be $q_0,q_1,q_2$.

Here, $q_0$ is start state and final state.

From the diagram:

$q_0\xrightarrow{a}q_1$

$q_1\xrightarrow{b}q_0$

$q_1\xrightarrow{a}q_2$

$q_2\xrightarrow{b}q_1$

$q_0\xrightarrow{b}q_2$

Now, paths from $q_0$ back to $q_0$ are:

$q_0\xrightarrow{a}q_1\xrightarrow{b}q_0$, giving $ab$

Also,

$q_0\xrightarrow{b}q_2\xrightarrow{b}q_1\xrightarrow{b}q_0$, giving $bbb$

In general, from $q_1$ to $q_0$ we can have:

$(ab)^*b$

So, complete cycle from $q_0$ to $q_0$ is:

$(a+bb)(ab)^*b$

Since $q_0$ is final, these cycles can repeat any number of times.

Required regular expression is:

$((a+bb)(ab)^*b)^*$