19 research outputs found
Using Graph Theory to Derive Inequalities for the Bell Numbers
The Bell numbers count the number of different ways to partition a set of
elements while the graphical Bell numbers count the number of non-equivalent
partitions of the vertex set of a graph into stable sets. This relation between
graph theory and integer sequences has motivated us to study properties on the
average number of colors in the non-equivalent colorings of a graph to discover
new non trivial inequalities for the Bell numbers. Example are given to
illustrate our approach
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
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 and stability number . Our main result is that the
edges of a non-trivial extremal tree can be partitioned into stars,
each of size or , 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
Facet defining inequalities among graph invariants: The system GraPHedron
AbstractWe 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