An isoperimetric inequality for eigenvalues of the bi-harmonic operator

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator ${\mathrm{\Delta }}^2$ on a bounded smooth domain $\text{Om}$ in the Euclidean $n$-space ${\mathbf{R}}^n$ ($n\ge 2$) and then prove that the corresponding first non-zero eigenvalue ${\mathrm{{\rm Y}}}_1(\text{Om})$ admits the isoperimetric inequality of Szeg"o-Weinberger type: ${\mathrm{{\rm Y}}}_1(\text{Om})\le {\mathrm{{\rm Y}}}_1(B_{\text{Om}})$, where $B_{\text{Om}}$ is a ball in ${\mathbf{R}}^n$ with the same volume of $\text{Om}$. The isoperimetric inequality of Szeg"o-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators ${\mathrm{\Delta }}^{2m}$ ($m\ge 1$) on $\text{Om}$ is also exploited.