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

Abstract: 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 ${\bf 1}_{k\times \cdots \times k}$. 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 $-r-1\leq l \leq r$, and $k$ sufficiently large.