The Nullities of Signed Cycle-Spliced Graphs

Let η(Γ)and c(Γ)be the nullity and the cyclomatic number of a signed graph Γ.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 Γ,η(Γ)≤ c(Γ)+1 and the extremal graphs Γ with nullity c(Γ)+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 Γ,η(Γ)≠ c(Γ).Some properties on signed cycle-spliced graphs Γ withη(Γ)=c(Γ)-1 are explored,as well as a structural characterization on signed cycle-spliced bipartite graphs Γ satisfying η(Γ)=c(Γ)-1.