Graphs and networks · Grafos e redes
| English | Português |
|---|---|
| graph/ɡræf/ | gráfico |
| vertices/ˈvɜːtɪsiːz/ | vértices |
| edges/ˈedʒɪz/ | bordas |
| degree/dɪˈɡriː/ | grau |
| path/pæθ/ | caminho |
| cycle/ˈsaɪkl/ | ciclo |
| connected/kəˈnektɪd/ | conectado |
| tree/triː/ | árvore |
| weighted graph/ˈweɪtɪd ɡræf/ | grafo ponderado |
| shortest path/ˈʃɔːtɪst pæθ/ | caminho mais curto |
A connection model needs its assumptions
- A graph 图 has vertices 顶点 and · e 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 · caminho 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? · Em um grafo simples não orientado sem laços, quatro arestas encontram um vértice. Qual é o seu grau?
Its degree is 4, the number of incident edges under the stated graph convention. · Seu grau é 4, o número de arestas incidentes sob a convenção de grafo declarada.
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? · Uma árvore tem 4 vértices. Quantas arestas ela tem?
A finite tree with four vertices has n − 1 = 3 edges. This count depends on the graph actually being connected and acyclic. · Uma árvore finita com quatro vértices tem n − 1 = 3 arestas. Essa contagem depende do grafo ser efetivamente conexo e acíclico.
In a connected undirected network with strictly positive edge costs, why does a minimum-cost connected subgraph spanning all vertices contain no cycle? · Em uma rede não direcionada conectada com custos de aresta estritamente positivos, por que um subgrafo conectado de custo mínimo que abrange todos os vértices não contém ciclos?
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. · Remover qualquer aresta de ciclo preserva a conectividade e reduz o custo total porque seu peso é estritamente positivo. Assim, um subgrafo cíclico não pode ser mínimo sob essas suposições.
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. · Combine cada situação ao que seus vértices e arestas devem representar.
Deciding what the vertices represent is the modelling step, and it is where the marks are. · Decidir o que os vértices representam é o passo de modelagem, e é onde estão as notas.
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. · Dois desenhos que parecem diferentes devem ser grafos diferentes.
Different layouts can have the same labelled adjacency. Compare directions and weights as well when those are part of the specified graph. · Diferentes representações podem ter a mesma adjacência rotulada. Compare também as direções e pesos quando estes fizerem parte do grafo especificado.
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.