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Linear inequalities among graph invariants: using GraPHedron to uncover optimal relationships
, 2005
"... Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph nodes, 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 ..."
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Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph nodes, 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 nodes. It greatly helps conjecturing optimal linear inequalities, which are then hopefully proved for any node number. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities in the stability number and the link number are obtained. To this aim, a problem of Ore (1962) related to Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter.
Digenes: genetic algorithms to discover conjectures about directed and undirected graphs
"... Abstract. We present Digenes, a new discovery system that aims to help researchers in graph theory. While its main task is to find extremal graphs for a given (function of) invariants, it also provides some basic support in proof conception. This has already been proved to be very useful to find new ..."
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Abstract. We present Digenes, a new discovery system that aims to help researchers in graph theory. While its main task is to find extremal graphs for a given (function of) invariants, it also provides some basic support in proof conception. This has already been proved to be very useful to find new conjectures since the AutoGraphiX system of Caporossi and Hansen [8]. However, unlike existing systems, Digenes can be used both with directed or undirected graphs. In this paper, we present the principles and functionality of Digenes, describe the genetic algorithms that have been designed to achieve them, and give some computational results and open questions. This do arise some interesting questions regarding genetic algorithms design particular to this field, such as crossover definition.
Some History of the Development of Graffiti
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science 69: Graphs and Discovery
"... Abstract. This paper provides some history of the development of the conjecture ..."
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Abstract. This paper provides some history of the development of the conjecture