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 er−1r with the equality holds if and only if e=kr for some integer k and all ones forms a principal sub-tensor 1k×⋯×k. We also prove a stability result for general tensor A with e ones where e=kr+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 kr+l ones, for −r−1≤l≤r, and k sufficiently large.