Extremal Graph Theory heavily relies on exploring bounds and inequalities between graph invariants, a task complicated by the rapid combinatorial explosion of graphs. Various tools have been developed to assist researchers in navigating this complexity, yet they typically rely on heuristic, probabilistic, or non-exhaustive methods, trading exactness for scalability. PHOEG takes a different stance: rather than approximating, it commits to an exact approach. PHOEG is an interactive online tool (https://phoeg.umons.ac.be) designed to assist researchers and educators in graph theory. Building upon the exact geometrical approach of its predecessor, GraPHedron, PHOEG embeds graphs into a two-dimensional invariant space and computes their convex hull, where facets represent inequalities and vertices correspond to extremal graphs. PHOEG modernizes and expands this approach by offering a comprehensive web interface and API, backed by an extensive database of pairwise non-isomorphic graphs including all graphs up to order 10. Users can intuitively define invariant spaces by selecting a pair of invariants, apply constraints and colorations, visualize resulting convex polytopes, and seamlessly inspect the corresponding drawn graphs. In this paper, we detail the software architecture and new web-based features of PHOEG. Furthermore, we demonstrate its practical value in two primary contexts: in research, by illustrating its ability to quickly identify conjectures or counterexamples to conjectures, and in education, by detailing its integration into university-level coursework to foster student discovery of classical graph theory principles. Finally, this paper serves as a brief survey of the extremal results and conjectures established over the past two decades using this geometric approach.

翻译：极值图论高度依赖于探索图不变量的界限与不等式，而图的组合爆炸使得这一任务异常复杂。目前已开发出多种工具帮助研究人员应对这一复杂性，但通常依赖启发式、概率性或非穷举方法，以牺牲准确性换取可扩展性。PHOEG采取了不同立场：它不追求近似，而致力于精确方法。PHOEG是一个交互式在线工具(https://phoeg.umons.ac.be)，旨在为图论研究人员和教育工作者提供支持。基于其前身GraPHedron的精确几何方法，PHOEG将图嵌入二维不变量空间并计算其凸包，其中面对应不等式，顶点对应极图。PHOEG通过提供全面的网络界面和API，并依托包含所有10阶以下图的非同构图数据库，对该方法进行了现代化和扩展。用户可以直观地选择一对不变量定义不变量空间，应用约束条件和着色方案，可视化生成的凸多面体，并无缝查看对应的绘制图。本文详述了PHOEG的软件架构和基于Web的新功能。此外，我们分两个主要场景展示其实用价值：在研究方面，说明其快速识别猜想或反例的能力；在教学方面，详述其如何融入大学课程以引导学生自主发现经典图论原理。最后，本文简要综述了过去二十年基于该几何方法建立的极值结果与猜想。