Theoretical study characterizes nonsingular structures in cycle-spliced signed graphs, highlighting nullity bounds for bipartite systems and odd cycles.
We consider the adjacency spectrum of cycle-spliced signed graphs (CSSG), i.e., signed graphs whose blocks are (independent) signed cycles. For a signed graph Σ, the nullity η(Σ) is the multiplicity of the 0-eigenvalue. The adjancency spectrum of cycle-spliced (signed) graphs is studied in the literature for the relation between the nullity η and the cyclomatic number c, in particular, it is known that 0≤η(Σ) ≤ c(Σ)+1. In this paper, nonsingular cycle-spliced bipartite signed graphs are characterized. For cycle-spliced signed graphs Σ having only odd cycles, we show that η(Σ) is 0 or 1. Finally, we compute the nullity of CSSGs consisting of at most three cycles.
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Khan et al. (2024) studied this question.
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