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Balance status of -G and -|G| when G is balanced

Determine whether, for a balanced signed graph G (i.e., κ(G)=1), the sign-flipped graph −G and the fully negative graph −|G| are themselves structurally balanced; equivalently, ascertain if κ(−G)=1 and κ(−|G|)=1 hold under the assumption that G is balanced.

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Background

In analyzing the relationship between global balance and conditioning, the paper considers related graphs obtained from G: the sign-flipped graph −G and the fully negative version −|G|. It notes that if G is balanced (κ(G)=1), then −G and −|G| are antibalanced; however, the authors state they do not know whether these graphs are balanced.

The authors can only assert κ(−G)≤1 and κ(−|G|)≤1, leaving open whether equality can occur in general or under specific conditions.

Clarifying the balance status of −G and −|G| when G is balanced would refine the structural implications of balance transformations and tighten bounds for κ on these derived graphs.

References

First, observe that if $G$ is balanced, namely $\kappa(G)=1$, both $-G$ and $-|G|$ are antibalanced. Actually, we do not know whether they are balanced or not, that is, we can only say that $\kappa(-G)\leq1$ and $\kappa(-|G|)\leq1$.

— Global Balance and Systemic Risk in Financial Correlation Networks (2407.14272 - Bartesaghi et al., 19 Jul 2024) in Section 4.2 (Global balance and condition number), preceding Proposition 3