AI 中文总结
该研究否证了正p-能量在p>=3时保持边添加单调性的猜想，证明对任意实数p>=1，添加边可使能量减小，构造基于团块链与正则二部图。
AI 中文摘要
在2021年AIM研讨会上，Guo猜想正平方能量s+ = E+_2应继承谱半径的熟悉的边添加单调性，即rho(G + uv) >= rho(G)。该猜想随后被证明在p = 2时失效。Tang、Liu和Wang随后引入了正p-能量，证明了对于每个1 <= p < 3的非单调性，并在其预印本第3版（2025年3月26日）中明确猜想对于p >= 3单调性应成立。我们完全否定了这一猜想的高指数扩展：对于每个实数p > 2，存在无穷多个连通图G和非边uv，使得E+_p(G + uv) < E+_p(G)。结合Tang-Liu-Wang在3以下的反例，这表明没有任何指数p >= 1能恢复谱半径式的单调性：对于每个实数p >= 1，正p-能量在添加一条边后可能减小。该构造是由正则二部图连接的一系列团块组成的链。一个等商收敛到Q = I + cA(P_k)。对于非整数p，对I - cA(P_k)的分数幂的二项式级数符号论证给出了所需的负端点项。在整数指数时，选择c跨越第一谱阈值恰好留下一个负特征值，而路径局部性迫使正谱贡献具有负号。我们还给出了p = 4时的一个完全有理的38顶点证书，并确定了p = 3时一个固定17顶点反例的完全失效区间。
英文摘要
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.
