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On price of symmetrisation
[article]

2013
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arXiv
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pre-print

Acknowledgements The authors would like to thank Alain Hertz, Gilles Caporossi and

arXiv:1310.2775v1
fatcat:vz2bkevr4bcejb5kl5tjlxacme
*Hadrien*Lepousé for useful discussions about Conjecture 9. ...##
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PHOEG Helps Obtaining Extremal Graphs
[article]

2017
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arXiv
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pre-print

In 2008,

arXiv:1712.07861v1
fatcat:lfyueokggzhytlieq6kgspoqie
*Mélot*presented GraPHedron [7] . This tool differs from the previous ones by its ideas. ...##
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Fibonacci Index and Stability Number of Graphs: a Polyhedral Study
[article]

2008
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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

arXiv:0811.1449v1
fatcat:6npw4h6tfzhqxoeluesdm73f6e
## 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.##
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Digenes: genetic algorithms to discover conjectures about directed and undirected graphs
[article]

2013
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arXiv
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pre-print

Moreover, Hansen and

arXiv:1304.7993v1
fatcat:bxifiewz4jaz3lzxrkis3i3ire
*Mélot*[26] proved that irregularity is maximal if and only if Table 1 denotes statistics regarding these three problems. ...##
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The average size of maximal matchings in graphs
[article]

2022
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arXiv
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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

arXiv:2204.10236v1
fatcat:ez5kb4swzfa5jdoneqairqzs5q
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... 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.##
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Maximum Eccentric Connectivity Index for Graphs with Given Diameter
[article]

2018
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arXiv
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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,

arXiv:1808.10203v1
fatcat:zsdu7vuvnrca3jptggq4f5gbia
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... e also characterize those graphs which have the largest eccentric connectivity index among all connected graphs of a given order n.##
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Integral complete split graphs

2002
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Publikacija Elektrotehnickog fakulteta - serija matematika
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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
##
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Facet defining inequalities among graph invariants: The system GraPHedron

2008
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Discrete Applied Mathematics
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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

doi:10.1016/j.dam.2007.09.005
fatcat:z4m6o7vydbhxrjhhfylgn2e2i4
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... 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.##
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Computers and discovery in algebraic graph theory

2002
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Linear Algebra and its Applications
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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
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Counting the number of non-equivalent vertex colorings of a graph

2016
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Discrete Applied Mathematics
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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
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A Tight Analysis of the Maximal Matching Heuristic
[chapter]

2005
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Lecture Notes in Computer Science
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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

doi:10.1007/11533719_71
fatcat:jch3ymqmgjaplhgfzjxwnaxba4
## 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.##
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Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pending Vertices
[article]

2018
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arXiv
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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

arXiv:1809.03158v1
fatcat:ytw5foyuy5dend2a4vzskgfuye
## 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.##
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Fibonacci index and stability number of graphs: a polyhedral study

2009
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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

doi:10.1007/s10878-009-9228-7
fatcat:mzfdrb3i3zbdfei2exegfxbnpm
## 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.##
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Using Graph Theory to Derive Inequalities for the Bell Numbers
[article]

2021
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arXiv
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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
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Upper bounds on the average number of colors in the non-equivalent colorings of a graph
[article]

2021
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arXiv
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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
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