House of Graphs: a database of interesting graphs

In this note we present House of Graphs (http://hog.grinvin.org) which is a new database of graphs. The key principle is to have a searchable database and offer -- next to complete lists of some graph classes -- also a list of special graphs that already turned out to be interesting and relevant in the study of graph theoretic problems or as counterexamples to conjectures. This list can be extended by users of the database.Comment: 8 pages; added a figur

Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate

Automated conjecturing III : property-relations conjectures

Discovery in mathematics is a prototypical intelligent behavior, and an early and continuing goal of artificial intelligence research. We present a heuristic for producing mathematical conjectures of a certain typical form and demonstrate its utility. Our program conjectures relations that hold between properties of objects (property-relation conjectures). These objects can be of a wide variety of types. The statements are true for all objects known to the program, and are the simplest statements which are true of all these objects. The examples here include new conjectures for the hamiltonicity of a graph, a well-studied property of graphs. While our motivation and experiments have been to produce mathematical conjectures-and to contribute to mathematical research-other kinds of interesting property-relation conjectures can be imagined, and this research may be more generally applicable to the development of intelligent machinery

Linear inequalities among graph invariants: Using GraPHedron to uncover optimal relationships

Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a "small" number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter. © 2008 Wiley Periodicals, Inc

Advancements in Research Mathematics through AI: A Framework for Conjecturing

This paper introduces a general framework for computer-based conjecture generation, particularly those conjectures that mathematicians might deem substantial and elegant. We describe our approach and demonstrate its effectiveness by providing examples of its application in producing publishable research and unexpected mathematical insights. We anticipate that our discussion of computer-assisted mathematical conjecturing will catalyze further research into this area and encourage the development of more advanced techniques than the ones presented herein

Techniques pour l'exploration de données structurées et pour la découverte de connaissances en théorie des graphes

Improving frequent subgraph mining in the presence of symmetry -- Using background knowledge to improve structured data mining -- Automated generation of conjectures on forbidden subgraph characterization