The kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs

Abstract

In this article, we study the kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs. A bound on the kth upper bases for primitive nonpowerful minimally strong signed digraphs is obtained, and the equality case of the bound is characterized. For the kth lower bases, we obtain some bounds. For some cases, the bounds are best possible and the extremal signed digraphs are characterized. We also show that there exist ‘gaps’ in both the kth upper base set and the kth lower base set of primitive nonpowerful minimally strong signed digraphs.

Keywords:

• base
• sign pattern
• digraph
• primitive matrix
• nonpowerful

AMS Subject Classifications::

• 05C20
• 05C50
• 15A48

Acknowledgements

The authors are grateful to the referee for many valuable comments and suggestions. This article was completed when the first author, Yanling Shao, was visiting Middle Tennessee State University. She is very grateful to Dr Rong Luo and the Department of Mathematical Sciences at Middle Tennessee State University for the invitation and hospitality to do this research. Shao and Gao's research is partially supported by NNSF of China (No. 11071227) and NSF of Shanxi (No. 2008011009). Shen's research is partially supported by NSF (CNS 0835834, 0934028, DMS 1005206) and Texas Higher Education Coordinating Board (ARP 003615-0039-2007).