Graphs and networks · Graphes et réseaux
| English | Français |
|---|---|
| graph/ɡræf/ | graphe |
| vertices/ˈvɜːtɪsiːz/ | sommet |
| edges/ˈedʒɪz/ | bords |
| degree/dɪˈɡriː/ | degré |
| path/pæθ/ | chemin |
| cycle/ˈsaɪkl/ | cycle |
| connected/kəˈnektɪd/ | connecté |
| tree/triː/ | arbre |
| weighted graph/ˈweɪtɪd ɡræf/ | graphe pondéré |
| shortest path/ˈʃɔːtɪst pæθ/ | chemin le plus court |
A connection model needs its assumptions
- A graph 图 has vertices 顶点 and · et edges 边. Decide what they represent and whether the connections are directed, weighted or undirected.
- The worksheet uses finite simple undirected graphs without loops or repeated edges. Drawing position does not change their connections, but an arrow or edge weight can change the mathematical model.
Count connections consistently
- Degree 度 counts incident edges in a simple undirected graph. The total degree is twice the number of edges, so it must be even.
- A path · chemin 路径 follows connections; a cycle 环 returns to its start without repeating other vertices. A graph is connected 连通 when each vertex can be reached from every other.
In a simple undirected graph without loops, four edges meet a vertex. What is its degree? · Dans un graphe simple non orienté sans boucles, quatre arêtes incident à un sommet. Quelle est sa degree (degré) ?
Its degree is 4, the number of incident edges under the stated graph convention. · Son degré est 4, le nombre d'arêtes incidentes selon la convention de graphe déclarée.
A tree needs two properties
- A tree 树 is connected and has no cycles. A finite tree with n vertices has n minus 1 edges.
- Having n minus 1 edges alone does not prove a tree: a triangle plus a separate two-vertex component has five vertices and four edges but is disconnected and cyclic.
A tree has 4 vertices. How many edges does it have? · Un arbre a 4 sommets. Combien d'arêtes a-t-il ?
A finite tree with four vertices has n − 1 = 3 edges. This count depends on the graph actually being connected and acyclic. · Un arbre fini à quatre sommets a n − 1 = 3 arêtes. Ce comptage dépend du fait que le graphe soit effectivement connecté et acyclique.
In a connected undirected network with strictly positive edge costs, why does a minimum-cost connected subgraph spanning all vertices contain no cycle? · Dans un réseau connexe non orienté avec des coûts d'arête strictement positifs, pourquoi un sous-graphe de coût minimal connectant tous les sommets ne contient-il aucun cycle ?
Removing any cycle edge preserves connectivity and lowers total cost because its weight is strictly positive. Thus a cyclic subgraph cannot be minimum under these assumptions. · Retirer une arête de cycle préserve la connectivité et réduit le coût total car son poids est strictement positif. Ainsi, un sous-graphe cyclique ne peut être minimal sous ces hypothèses.
Positive-cost connecting roads. For a connected network with strictly positive road costs, removing a cycle edge leaves connectivity and lowers cost. A minimum-cost connecting subgraph is therefore a tree. To find a minimum spanning tree, choose edges that connect all vertices without a cycle; blindly choosing the cheapest n minus 1 edges can leave separate components.
A shortest route is not a spanning tree
- A weighted graph 带权图 attaches a specified value to each edge. A shortest path 最短路径 minimises route weight between the given endpoints.
- A minimum spanning tree instead connects every vertex at minimum total included weight. A shortest route can have more edges than a direct expensive route; sum the weights rather than count the edges.
Match each situation to what its vertices and edges should be. · Reliez chaque situation à ce que ses sommets et arêtes devraient représenter.
Deciding what the vertices represent is the modelling step, and it is where the marks are. · Décider ce que représentent les sommets est l'étape de modélisation, et c'est là que se trouvent les points.
A closed edge changes feasibility. Sheet 4.5 finds a route of weight 7, then closes an edge and checks a replacement of weight 9. Do not reuse the old total for a path that no longer exists.
Two drawings that look different must be different graphs. · Deux dessins différents doivent être considérés comme différents graphes.
Different layouts can have the same labelled adjacency. Compare directions and weights as well when those are part of the specified graph. · Des mises en page différentes peuvent avoir la même adjacence étiquetée. Comparez également les directions et les poids lorsque ceux-ci font partie du graphe spécifié.
Compare all structure relevant to the question: vertex correspondence, adjacency, directions and weights where specified. Equal-looking pictures need not represent the same graph; different layouts can represent the same labelled connections.