Spectral Radius of $\{0,1\}$-Tensor with Prescribed Number of Ones

For any $r$-order $\{0,1\}$-tensor $A$ with $e$ ones, we prove that the spectral radius of $A$ is at most $e^{\frac{r-1}{r}}$ with the equality holds if and only if $e=k^r$ for some integer $k$ and all ones forms a principal sub-tensor ${\mathbf{1}}_{k\times \cdots \times k}$. We also prove a stability result for general tensor $A$ with $e$ ones where $e=k^r+l$ with relatively small $l$. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all $r$-order $\{0,1\}$-tensor $A$ with $k^r+l$ ones, for $-r-1\le l\le r$, and $k$ sufficiently large.