Gauvain Devillez

Gauvain Devillez, Sven D'hondt, Jan Goedgebeur

The House of Graphs is an online database of graphs which can be accessed at https://houseofgraphs.org/. It serves as a central repository for complete lists of graphs for various graph classes. However, its main feature is a searchable database of so-called "interesting" graphs. The development of the original House of Graphs started in 2010 and it was completely rebuilt in 2021-2022. Each graph in the database is accompanied by a significant amount of meta-data such as a name, drawings, precomputed graph invariants, and comments. Given this volume of information and the importance of reliability in the scientific world, robust data management is essential to ensure accuracy and consistency across the database. In this article, we therefore focus on knowledge management in the House of Graphs and describe the inner workings of the House of Graphs and how we ensure that its data is coherent, qualitative and stable.

Sébastien Bonte, Gauvain Devillez, Valentin Dusollier et al.

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.