Filters








41 Hits in 0.97 sec

On price of symmetrisation [article]

Absil Romain, Hadrien Mélot
2013 arXiv   pre-print
Acknowledgements The authors would like to thank Alain Hertz, Gilles Caporossi and Hadrien Lepousé for useful discussions about Conjecture 9.  ... 
arXiv:1310.2775v1 fatcat:vz2bkevr4bcejb5kl5tjlxacme

PHOEG Helps Obtaining Extremal Graphs [article]

Gauvain Devillez, Pierre Hauweele, Hadrien Mélot
2017 arXiv   pre-print
In 2008, Mélot presented GraPHedron [7] . This tool differs from the previous ones by its ideas.  ... 
arXiv:1712.07861v1 fatcat:lfyueokggzhytlieq6kgspoqie

Fibonacci Index and Stability Number of Graphs: a Polyhedral Study [article]

Véronique Bruyère, Hadrien Mélot
2008 arXiv   pre-print
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
more » ... y 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.
arXiv:0811.1449v1 fatcat:6npw4h6tfzhqxoeluesdm73f6e

Digenes: genetic algorithms to discover conjectures about directed and undirected graphs [article]

Romain Absil, Hadrien Mélot
2013 arXiv   pre-print
Moreover, Hansen and Mélot [26] proved that irregularity is maximal if and only if Table 1 denotes statistics regarding these three problems.  ... 
arXiv:1304.7993v1 fatcat:bxifiewz4jaz3lzxrkis3i3ire

The average size of maximal matchings in graphs [article]

Alain Hertz and Sébastien Bonte and Gauvain Devillez and Hadrien Mélot
2022 arXiv   pre-print
We investigate the ratio I(G) of the average size of a maximal matching to the size of a maximum matching in a graph G. If many maximal matchings have a size close to μ(G), this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, I(G) approaches 1/2. We propose a general technique to determine the asymptotic behavior of (G) for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known
more » ... ptotic values of I(G) which were typically obtained using generating functions, and we then determine the asymptotic value of I(G) for other families of graphs, highlighting the spectrum of possible values of this graph invariant between 1/2 and 1.
arXiv:2204.10236v1 fatcat:ez5kb4swzfa5jdoneqairqzs5q

Maximum Eccentric Connectivity Index for Graphs with Given Diameter [article]

Pierre Hauweele, Alain Hertz, Hadrien Mélot, Bernard Ries and Gauvain Devillez
2018 arXiv   pre-print
The eccentricity of a vertex v in a graph G is the maximum distance between v and any other vertex of G. The diameter of a graph G is the maximum eccentricity of a vertex in G. The eccentric connectivity index of a connected graph is the sum over all vertices of the product between eccentricity and degree. Given two integers n and D with D≤ n-1, we characterize those graphs which have the largest eccentric connectivity index among all connected graphs of order n and diameter D. As a corollary,
more » ... e also characterize those graphs which have the largest eccentric connectivity index among all connected graphs of a given order n.
arXiv:1808.10203v1 fatcat:zsdu7vuvnrca3jptggq4f5gbia

Integral complete split graphs

Pierre Hansen, Hadrien Melot, Dragan Stevanovic
2002 Publikacija Elektrotehnickog fakulteta - serija matematika  
We give characterizations of integral graphs in the family of complete split graphs and a few related families of graphs.
doi:10.2298/petf0213089h fatcat:5kehyk6hyrafhdz2jwburpnfgy

Facet defining inequalities among graph invariants: The system GraPHedron

Hadrien Mélot
2008 Discrete Applied Mathematics  
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
more » ... 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.
doi:10.1016/j.dam.2007.09.005 fatcat:z4m6o7vydbhxrjhhfylgn2e2i4

Computers and discovery in algebraic graph theory

Pierre Hansen, Hadrien Mélot
2002 Linear Algebra and its Applications  
We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. Résumé On passe en revue les systèmes informatiques qui aident à obtenir et parfois donnent automatiquement des conjectures et réfutations en théorie algébrique des graphes.
doi:10.1016/s0024-3795(02)00386-5 fatcat:qxtwo74ygra5la6vj2fpscn2mq

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

Alain Hertz, Hadrien Mélot
2016 Discrete Applied Mathematics  
We study the number P(G) of non-equivalent ways of coloring a given graph G, also known as the (graphical) Bell number of G. We show some similarities and differences between this graph invariant and the well known chromatic polynomial. We then relate P(G) to Stirling numbers of the second kind, and to Bell, Fibonacci, and Lucas numbers, by computing the values of this invariant for some families of graphs. We finally study upper and lower bounds on P(G) for graphs with fixed maximum degree.
doi:10.1016/j.dam.2015.09.015 fatcat:pbp5qasvsjagpm4rd5j6v4bagu

A Tight Analysis of the Maximal Matching Heuristic [chapter]

Jean Cardinal, Martine Labbé, Stefan Langerman, Eythan Levy, Hadrien Mélot
2005 Lecture Notes in Computer Science  
We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a well-known 2-approximation to three classical NP-hard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. We show that the worst-case approximation factor of this simple method can be expressed in a finer way when assumptions on the density of the graph is made. For graphs with an average degree at least
more » ... called weakly -dense graphs, we show that the asymptotic approximation factor is min{2, 1/(1 − √ 1 − )}. For graphs with a minimum degree at least n -strongly -dense graphs -we show that the asymptotic approximation factor is min{2, 1/ }. These bounds are obtained through a careful analysis of the tight examples. Chercheur qualifié du F.N.R.S. Corresponding author.
doi:10.1007/11533719_71 fatcat:jch3ymqmgjaplhgfzjxwnaxba4

Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pending Vertices [article]

Gauvain Devillez, Alain Hertz, Hadrien Mélot, Pierre Hauweele
2018 arXiv   pre-print
The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product d_G(v) e_G(v), where d_G(v) is the degree of v in G and e_G(v) is the maximum distance between v and any other vertex of G. This index is helpful for the prediction of biological activities of diverse nature, a molecule being modeled as a graph where atoms are represented by vertices and chemical bonds by edges. We characterize those graphs which have the smallest eccentric connectivity index
more » ... ong all connected graphs of a given order n. Also, given two integers n and p with p≤ n-1, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order n with p pending vertices.
arXiv:1809.03158v1 fatcat:ytw5foyuy5dend2a4vzskgfuye

Fibonacci index and stability number of graphs: a polyhedral study

Véronique Bruyère, Hadrien Mélot
2009 Journal of combinatorial optimization  
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
more » ... y 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.
doi:10.1007/s10878-009-9228-7 fatcat:mzfdrb3i3zbdfei2exegfxbnpm

Using Graph Theory to Derive Inequalities for the Bell Numbers [article]

Alain Hertz and Anaelle Hertz and Hadrien Mélot
2021 arXiv   pre-print
The Bell numbers count the number of different ways to partition a set of n 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.
arXiv:2104.00552v2 fatcat:k3vizauhqzgmxmvyibghef472y

Upper bounds on the average number of colors in the non-equivalent colorings of a graph [article]

Alain Hertz, Hadrien Mélot, Sébastien Bonte, Gauvain Devillez, Pierre Hauweele
2021 arXiv   pre-print
A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let 𝒜(G) be the average number of colors in the non-equivalent colorings of a graph G. We give a general upper bound on 𝒜(G) that is valid for all graphs G and a more precise one for graphs G of order n and maximum degree Δ(G)∈{1,2,n-2}.
arXiv:2105.01120v1 fatcat:5umi6mmjxnfunmxa67t3ejj4uu
« Previous Showing results 1 — 15 out of 41 results