For connected claw-free graphs with maximum degree at least 3 and for diameter-2 graphs other than stars and C5, the positive square energy is at least the number of vertices.
Positive and negative square energies of graphs.Electron
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Refinement of a conjecture on positive square energy of graphs
For connected claw-free graphs with maximum degree at least 3 and for diameter-2 graphs other than stars and C5, the positive square energy is at least the number of vertices.