可积图的新族与运算
New Families and Operations on Integrable Graphs
AI总结：
本文定义了辐条图与反克隆算子，证明辐条图生成的动力学可积，反克隆算子作用于奇数顶点平衡竞赛图生成可积图，作用于有向环则产生混沌，完善了6顶点内有向图可积动力学分类。
AI中文摘要：
本文研究由对应图结构生成的相互作用矩阵所产生的Lotka-Volterra（复制子）系统的可积性，延续了Visomirski和Griffin[J. Phys. A., 58:015701, 2025]以及Evripidou等人[J. Phys. A., 55:325201, 2022]等学者的工作。特别地，我们定义了一类新的图——辐条图（spoked graphs），并证明由该类图生成的所有动力学均为可积的。针对Evripidou等人（2022）的研究，我们定义了一种新的反克隆算子（anti-cloning operator），并证明该算子作用于平衡竞赛图（顶点数为奇数）时，会生成动力学可积的新图。有趣的是，我们提供了数值证据表明，当将该反克隆运算应用于其他图族（例如生成经典可积Volterra格的有向环）时，会导致混沌行为。本研究完善了Visomirski和Griffin（2025）启动的、由顶点数不超过6的有向图生成的所有可积动力学的分类，并为该主题提出了若干未来研究方向。
英文摘要：
In this paper, we investigate the integrability of Lotka-Volterra (replicator) systems arising from interaction matrices generated from corresponding graph structures, continuing work started by Visomirski and Griffin [J. Phys. A., 58:015701, 2025] and Evripidou et al. [J. Phys. A., 55:325201, 2022] (among others). In particular, we define a new family of graphs, the spoked graphs, and show that all dynamics generated from this family are integrable. In reference to Evripidou et al. (2022), we define a new anti-cloning operator and show that its action on balanced tournament graphs (with odd vertex count) generates new graphs whose dynamics are integrable. Interestingly, we provide numerical evidence that this anti-cloning operation leads to chaotic behaviour when applied to other graph families (e.g., the directed cycles that generate the classically integrable Volterra lattice). This work completes a taxonomy of all integrable dynamics generated by directed graphs with up to six vertices started by Visomirski and Griffin (2025), and suggests several future directions of study on this topic.
