17 research outputs found
Proximity and Remoteness in Graphs: a survey
The proximity of a connected graph is the minimum, over
all vertices, of the average distance from a vertex to all others. Similarly,
the maximum is called the remoteness and denoted by . The
concepts of proximity and remoteness, first defined in 2006, attracted the
attention of several researchers in Graph Theory. Their investigation led to a
considerable number of publications. In this paper, we present a survey of the
research work.Comment: arXiv admin note: substantial text overlap with arXiv:1204.1184 by
other author
On the Nullity Number of Graphs
The paper discusses bounds on the nullity number of graphs. It is proved in [B. Cheng and B. Liu, On the nullity of graphs. Electron. J. Linear Algebra 16 (2007) 60--67] that , where , n and D denote the nullity number, the order and the diameter of a connected graph, respectively. We first give a necessary condition on the extremal graphs corresponding to that bound, and then we strengthen the bound itself using the maximum clique number. In addition, we prove bounds on the nullity using the number of pendant neighbors in a graph. One of those bounds is an improvement of a known bound involving the domination number
Comparaison automatisée d'invariants en théorie des graphes
Intelligence artificielle et découverte automatisée -- De autographix à autographix 2 -- Comparaison systématique de quelques invariants
Some properties of the distance Laplacian eigenvalues of a graph
summary:The distance Laplacian of a connected graph is defined by , where is the distance matrix of , and is the diagonal matrix whose main entries are the vertex transmissions in . The spectrum of is called the distance Laplacian spectrum of . In the present paper, we investigate some particular distance Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance Laplacian eigenvalues. We establish some properties of the distance Laplacian spectrum that enable us to derive the distance Laplacian characteristic polynomials for several classes of graphs
Recherche à voisinage variable de graphes extrémaux 13. à propos de la maille
Le système AutoGraphiX (AGX1 et AGX2) permet,
parmi d'autres fonctions, la génération automatique de conjectures en
théorie des graphes et, dans une version plus récente, la preuve automatique de conjectures simples. Afin
d'illustrer ces fonctions et le type de résultats obtenus, nous étudions systématiquement ici des conjectures
obtenues par ce système et de la forme où g désigne la maille (ou longueur
du plus petit cycle) du graphe G=(V, E), i un autre invariant choisi
parmi le nombre de stabilité, le rayon, le diamètre, le degré minimum,
moyen ou maximum, et des fonctions de
l'ordre n = |V| de G les meilleures possibles, enfin correspond
à une des opérations +,-,×,/.
48 telles conjectures sont obtenues: les plus simples sont démontrées
automatiquement et les autres à la main. De plus 12 autres conjectures
ouvertes et non encore étudiées sont soumises aux lecteurs
Variable neighborhood search for extremal graphs. 17. Further conjectures and results about the index
The AutoGraphiX 2 system is used to compare the index of a connected graph G with a number of other graph theoretical invariants, i.e., chromatic number, maximum, minimum and average degree, diameter, radius, average distance, independence and domination numbers. In each case, best possible lower and upper bounds, in terms of the order of G, are sought for sums, differences, ratios and products of the index and another invariant. There are 72 cases altogether: in 7 cases known results were reproduced, in 32 cases immediate results were obtained and automatically proved by the system, conjectures were obtained in 27 cases, of which 12 were proved (in 3 theorems and 9 propositions), 9 remain open and 6 were refuted. No results could be derived in 7 cases
Variable Neighborhood Search for Extremal Graphs. 21. Conjectures
A set of vertices S in a graph G is independent if no neighbor of a vertex of S belongs to S. The independence number alpha is the maximum cardinality of ail independent set of G. A series of best possible lower and Upper bounds on alpha, and some other common invariants of G are obtained by the system AGX 2. and proved either automatically or by hand. In the present paper, we report on such lower and upper bounds considering, as second invariant, minimum, average and maximum degree, diameter, radius, average distance, spread of eccentricities, chromatic number and matching number