In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.
In the case of a directed graph the distance d(u,v) between two vertices u and v is defined as the length of a shortest directed path from u to v consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, d(u,v) does not necessarily coincide with d(v,u)—so it is just a quasi-metric, and it might be the case that one is defined while the other is not.
A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric.
The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected.
The eccentricity ε(v) of a vertex v is the greatest distance between v and any other vertex; in symbols,
It can be thought of as how far a node is from the node most distant from it in the graph.
The radius r of a graph is the minimum eccentricity of any vertex or, in symbols,
The diameter d of a graph is the maximum eccentricity of any vertex in the graph. That is, d is the greatest distance between any pair of vertices or, alternatively,
To find the diameter of a graph, first find the shortest path between each pair of vertices. The greatest length of any of these paths is the diameter of the graph.
A central vertex in a graph of radius r is one whose eccentricity is r—that is, a vertex whose distance from its furthest vertex is equal to the radius, equivalently, a vertex v such that ε(v) = r.
A peripheral vertex in a graph of diameter d is one whose eccentricity is d—that is, a vertex whose distance from its furthest vertex is equal to the diameter.
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Study of structures and concepts that do not require the notion of continuity. Graph theory, or study of general countable sets are some of the areas that are covered by discrete mathematics. Emphasis
Internet analytics is the collection, modeling, and analysis of user data in large-scale online services, such as social networking, e-commerce, search, and advertisement. This class explores a number
A first graduate course in algorithms, this course assumes minimal background, but moves rapidly. The objective is to learn the main techniques of algorithm analysis and design, while building a reper
thumb|upright=3|Les graphes en étoile S3, S4, S5 et S6. En mathématiques, et plus particulièrement en théorie des graphes, une étoile Sk est le graphe biparti complet K1,k. On peut aussi le voir comme un arbre avec un nœud et k feuilles, du moins lorsque k > 1. Enfin, on peut le définir comme un graphe connexe dont tous les sommets sauf un sont de degré 1. Certains auteurs définissent toutefois Sk comme l'arbre à k sommets de diamètre maximal 2. Attention, avec cette définition, une étoile n'a que k − 1 feuilles.
En sciences humaines et sociales, l'expression réseau social désigne un agencement de liens entre des individus ou des organisations, constituant un groupement qui a un sens : la famille, les collègues, un groupe d'amis, une communauté, etc. L'anthropologue australien John Arundel Barnes a introduit l'expression en 1954. L'analyse des réseaux sociaux est devenue une spécialité universitaire dans le champ de la sociologie, se fondant sur la théorie des réseaux et l'usage des graphes.
En théorie des graphes, un invariant de graphe est une quantité qui n'est pas modifiée par isomorphisme de graphes. Un invariant de graphe ne dépend donc que de la structure abstraite et pas des particularités de la représentation comme l'étiquetage ou le tracé. De nombreux invariants sont conservés par certains préordres ou ordres partiels naturels sur les graphes : Une propriété est monotone si elle est héritée par les sous-graphes. Le caractère biparti, sans triangle, ou planaire sont des exemples de propriétés monotones.
Explore les distances sur les graphiques, les normes de coupe, les arbres de couverture, les modèles de blocs, les métriques, les normes et les ERGM dans l'analyse des données du réseau.
Water distribution systems (WDSs) are complex networks with numerous interconnected junctions and pipes. The robustness and reliability of these systems are critically dependent on their network structure, necessitating detailed analysis for proactive leak ...
Basel2024
This article focuses on spectral methods for recovering communities in temporal networks. In the case of fixed communities, spectral clustering on the simple time-aggregated graph (i.e., the weighted graph formed by the sum of the interactions over all tem ...
Ieee Computer Soc2024
,
Graph neural networks (GNNs) have demonstrated promising performance across various chemistry-related tasks. However, conventional graphs only model the pairwise connectivity in molecules, failing to adequately represent higher order connections, such as m ...