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

Abstract

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 <= p < 3, and in version 3 of their preprint (26 March 2025) explicitly conjectured that monotonicity should hold for p >= 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) < E+_p(G). Together with the Tang-Liu-Wang counterexamples below 3, this shows that no exponent p >= 1 restores the spectral-radius-style monotonicity: positive p-energy can decrease under the addition of an edge for every real p >= 1. The construction is a chain of clique blocks joined by regular bipartite graphs. An equitable quotient converges to Q = I + cA(P_k). For noninteger p, a binomial-series sign argument for a fractional power of I - cA(P_k) gives the required negative endpoint entry. At integer exponents, choosing c across the first spectral threshold leaves exactly one negative eigenvalue, and path locality forces the positive spectral contribution to have negative sign. We also give a fully rational 38-vertex certificate at p = 4 and determine the complete failure interval of a fixed 17-vertex counterexample at p = 3.