On the Hilbert depth of the quotient ring of the edge ideal of a star graph

Let $S_n=K[x_1,\dots ,x_n,y]$ and $I_n=(x_{1}y,x_{2}y,\dots ,x_{n}y)\subset S_n$ be the edge ideal of star graph. We prove that $\mathrm{hdepth}(S_n\mathrm{/}{I}_n)\ge \lceil \frac{n}{2}\rceil +\lfloor \sqrt{n}\rfloor -2$. Also, we show that for any $\epsilon >0$, there exists some integer $A=A(\epsilon )\ge 0$ such that $\mathrm{hdepth}(S_n\mathrm{/}{I}_n)\le \lceil \frac{n}{2}\rceil +\lfloor \epsilon n\rfloor +A-2$. We deduce that $\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\mathrm{hdepth}(S_n\mathrm{/}{I}_n)=\frac{1}{2}$.