The kth Upper Bases of Primitive Non-powerful Signed Digraphs

TL;DR

This paper investigates the properties of the kth upper bases in primitive non-powerful signed digraphs, establishing bounds, characterizing equality cases, and revealing gaps in the upper base set.

Contribution

It provides the first bounds on the kth upper bases of primitive non-powerful signed digraphs and characterizes when these bounds are tight.

Findings

Established bounds on the kth upper bases

Characterized equality cases for the bounds

Identified gaps in the upper base set

Abstract

In this paper, we study the kth upper bases of primitive non-powerful signed digraphs. A bound on the kth upper bases of all primitive non-powerful signed digraphs is obtained, and the equality case of the bound is characterized. We also show that there exists "gap" in the kth upper base set of primitive non-powerful signed digraphs.