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Research articles

ScienceAsia 50 (2024):ID 2024013 1-8 |doi: 10.2306/scienceasia1513-1874.2024.013


No T-gain graph with the rank r (ϕ) = 2m(G) - 2c(G) + 1


Yuxuan Wang, Rentian Shang, Jingwen Wu, Yong Lu*

 
ABSTRACT:     Let ? = (G,?) be a T-gain graph. In this paper, we will prove that there are no T-gain graphs with the rank 2m(G) ? 2c(G) + 1, where c(G) is the dimension of cycle space of G, m(G) is the matching number of G. For a given c(G), we also prove that there are infinitely many connected T-gain graphs with the rank 2m(G)?2c(G)+s, (0 ? s ? 3c(G),s ?= 1). These results can also apply to undirected graphs, signed graphs and mixed graphs.

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a School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116 China

* Corresponding author, E-mail: luyong@jsnu.edu.cn

Received 3 Aug 2022, Accepted 24 Nov 2023