The Nullities of Signed Cycle-Spliced Graphs
Received:December 14, 2022  Revised:April 25, 2023
Key Words: signed graph   nullity   cyclomatic number  
Fund Project:Supported by the Natural Science Foundation of Inner Mongolia Autonomous Region (Grant No.2020BS01011), the National Natural Science Foundation of China (Grant No.12171089), the Natural Science Foundation of Fujian Province (Grant No.2021J02048) and the Research Fund of Xiamen University of Technology (Grant Nos.YKJ20018R; XPDKT20039).
Author NameAffiliation
Sarula CHANG College of Science, Inner Mongolia Agricultural University, Inner Mongolia 010018, P. R. China 
Jianxi LI School of Mathematics and Statistics, Minnan Normal University, Fujian 363000, P. R. China 
Yirong ZHENG School of Mathematics and Statistics, Xiamen University of Technology, Fujian 361000, P. R. China 
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      Let $\eta(\Gamma)$ and $c(\Gamma)$ be the nullity and the cyclomatic number of a signed graph $\Gamma$. A signed cycle-spliced graph is a connected signed graph in which every block is a cycle. In this paper, we prove that for every signed cycle-spliced graph $\Gamma$, $\eta(\Gamma)\leq c(\Gamma)+1$ and the extremal graphs $\Gamma$ with nullity $c(\Gamma)+1$ are characterized, which extend the related results of Wong, Zhou and Tian (2022) on simple cycle-spliced graphs. Moreover, we prove that for every signed cycle-spliced graph $\Gamma$, $\eta(\Gamma)\neq c(\Gamma)$. Some properties on signed cycle-spliced graphs $\Gamma$ with $\eta(\Gamma)= c(\Gamma)-1$ are explored, as well as a structural characterization on signed cycle-spliced bipartite graphs $\Gamma$ satisfying $\eta(\Gamma)= c(\Gamma)-1$.
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