Graphs and networks
Introduced| English | Meaning |
|---|---|
| graph/ɡræf/ | A mathematical structure consisting of vertices connected by edges. |
| vertices/ˈvɜːtɪsiːz/ | The nodes of a graph; in geometry, the corner points where edges meet. |
| edges/ˈedʒɪz/ | Connections between vertices in a graph, or boundary segments of a geometric shape. |
| degree/dɪˈɡriː/ | An angular unit equal to one three-hundred-and-sixtieth of a full turn. |
| path/pæθ/ | path |
| cycle/ˈsaɪkl/ | cycle |
| connected/kəˈnektɪd/ | connected |
| tree/triː/ | A connected graph containing no cycles. |
| weighted graph/ˈweɪtɪd ɡræf/ | weighted graph |
| shortest path/ˈʃɔːtɪst pæθ/ | A path with the smallest total edge weight between specified vertices, or the fewest edges when all edges have equal weight. |
A connection model needs its assumptions
- A graph 图 has vertices 顶点 and 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 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?
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Its degree is 4, the number of incident edges under the stated graph convention.
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?
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A finite tree with four vertices has n − 1 = 3 edges. This count depends on the graph actually being connected and acyclic.
In a connected undirected network with strictly positive edge costs, why does a minimum-cost connected subgraph spanning all vertices contain no cycle?
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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.
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.
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Deciding what the vertices represent is the modelling step, and it is where the marks are.
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.
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Different layouts can have the same labelled adjacency. Compare directions and weights as well when those are part of the specified graph.
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.