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

TL;DR

This paper establishes bounds on the spectral radius of ,1tensors with a fixed number of ones, characterizes extremal tensors, and provides stability results for near-extremal cases.

Contribution

It introduces a precise upper bound for the spectral radius of ,1tensors with a given number of ones and characterizes the tensors that attain this maximum.

Findings

Spectral radius is at most $e^{rac{r-1}{r}}$ for tensors with $e$ ones.

Maximum spectral radius tensors are principal sub-tensors of all ones when $e=k^r$.

Stability results describe the spectral radius behavior for tensors close to extremal configurations.

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.