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arXiv · 1801.02784

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

Shuliang Bai·Linyuan Lu
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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.

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Shuliang Bai, Linyuan Lu. 2018-01-09. Spectral Radius of $\{0, 1\}$-Tensor with Prescribed Number of Ones. https://arxiv.org/abs/1801.02784

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