Expanding phenomena over matrix rings

Expanding phenomena over higher dimensional matrix rings

“…In this section, we mimic the study of sum-product digraph over M 2 (F q ) in [3] to extend results over M n (F q ). Let G be a directed graph (digraph) on n vertices where the in-degree and out-degree of each vertex are both d.…”

“…In [3], Y. D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author constructed some families of moderate expanders over M 2 (F q ).…”

“…In [3], Karabulut, Koh, Pham, Shen, and the second listed author obtained the following sum-product estimate over the matrix ring M 2 (F q ). Theorem 1.12.…”