Hadrien Mélot

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 have 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.

Facet defining inequalities among graph invariants: The system GraPHedron

We present a new computer system, called GraPHedron, which uses a polyhedral approach to help the user to discover optimal conjectures in graph theory. We define what should be optimal conjectures and propose a formal framework allowing to identify them. Here, graphs with n nodes are viewed as points in the Euclidian space, whose coordinates are the values of a set of graph invariants. To the convex hull of these points corresponds a finite set of linear inequalities. These inequalities computed for a few values of n can be possibly generalized automatically or interactively. They serve as conjectures which can be considered as optimal by geometrical arguments. We describe how the system works, and all optimal relations between the diameter and the number of edges of connected graphs are given, as an illustration. Other applications and results are mentioned, and the forms of the conjectures that can be currently obtained with GraPHedron are characterized.

A tight analysis of the maximal matching heuristic

We study the worst-case performance of the maximal matching heuristic applied to the Minimum Vertex Cover and Minimum Maximal Matching problems, through a careful analysis of tight examples. We show that the tight worst-case approximation ratio is asymptotic to

Computers and discovery in algebraic graph theory

{\rm min}\, \{2, 1/(1-\sqrt{1-\epsilon})\}

Fibonacci index and stability number of graphs: a polyhedral study

for graphs with an average degree at least en and to min {2, 1/e} for graphs with a minimum degree at least en.

Counting the number of non-equivalent vertex colorings of a graph

Abstract We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.

Turán 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. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a connected variant of them are also extremal for these particular problems. We also make a polyhedral study by establishing all the optimal linear inequalities for the stability number and the Fibonacci index, inside the classes of general and connected graphs of order n.

Trees with Given Stability Number and Minimum Number of Stable Sets

We give some extremal properties on the number B ( G ) of non-equivalent ways of coloring a given graph G , also known as the (graphical) Bell number of G . In particular, we study bounds on B ( G ) for graphs with a maximum degree constraint. First, an upper bound on B ( G ) is given for graphs with fixed order n and maximum degree Δ . Then, we give lower bounds on B ( G ) for fixed order n and maximum degree 1, 2, n - 2 and n - 1 . In each case, the bound is tight and we describe all graphs that reach the bound with equality.

A sharp lower bound on the number of non-equivalent colorings of graphs of order n and maximum degree n-3

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. Turan graphs frequently appear in extremal graph theory. We show that Turan graphs and a connected variant of them are also extremal for these particular problems.

Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

We study the structure of trees minimizing their number of stable sets for given order n and stability number α. Our main result is that the edges of a non-trivial extremal tree can be partitioned into n − α stars, each of size