0–1 matrices whose k-th powers have bounded entries

Extremal digraphs avoiding distinct walks of length 4 with the same endpoints

“…Extremal Digraphs Avoiding Distinct Walks of Length 4 with ... 3 We define H(n) as a family of digraphs of order n, each of whose elements has a vertex partition…”

“…The Turán-type problem is one of the hottest topics in extremal graph theory, which concerns the number of edges in graphs containing no given subgraphs and the extremal graphs achieving this maximum. Most of the previous results of Turán problems concern undirected graphs and only a few Turán problems on digraphs have been investigated; see [1][2][3][4][5][6].…”

Digraphs that contain at most t distinct walks of a given length with the same endpoints

“…Given two positive integers k, t, denote by F k,t the family of digraphs consisting of t different walks of length k with the same initial vertex and the same terminal vertex. In [5], the authors posed a Turán type problem as follows.…”

“…In the last decade, Problem 1 for the case t = 1 has been completely solved by Wu [12], by Huang and Zhan [8], by Huang, Lyu and Qiao [7], by Lyu [11], and by Huang and Lyu [6]. For the general cases of Problem 1, the case k = 2 has been studied in [9], and the case for k ≥ n − 1 ≥ 6t + 1 has been solved in [5].…”

On k-idempotent 0-1 matrices

“…Huang, Lyu, Qiao, Wu and Zhan [6,[8][9][10] investigated 0-1 matrices whose powers are also 0-1 matrices. Huang and Lyu [7] studied 0-1 matrices whose powers have bounded entries.…”