For any signed graph, the positive inertia index is at least (n - p)/2 minus the cyclomatic number, and equality holds exactly for disjoint unions of certain signed cycles.
Devadas Acharya, Spectral criterion for cycle balance in networks, Journal of Graph Theory, 4 (1) (1980) 1-11
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Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices
For any signed graph, the positive inertia index is at least (n - p)/2 minus the cyclomatic number, and equality holds exactly for disjoint unions of certain signed cycles.