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Edge-addition monotonicity of positive p-energy fails for every p >= 1

arXiv · AI, language, vision and robotics · article · Sep 11, 2026 · UTC

At a 2021 AIM workshop, Guo conjectured that the positive square energy s+ = E+_2 should inherit the familiar edge-addition monotonicity of the spectral radius, rho(G + uv) >= rho(G). That conjecture was subsequently shown to fail at p = 2. Tang, Liu, and Wang then introduced positive p-energy, proved nonmonotonicity for every 1 = 3. We disprove this conjectured high-exponent extension completely: for every real p > 2 there are infinitely many connected graphs G and nonedges uv such that E+_p(G + uv) = 1 restores the spectral-radius-style monotonicity: positive p-energy can decrease under the

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