Graphs and trees
Introduced| English | Meaning |
|---|---|
| graph/ɡræf/ | A mathematical structure consisting of vertices connected by edges. |
| edge/edʒ/ | 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. |
| directed | directed |
| vertex/ˈvɜːteks/ | A corner or meeting point of edges, a turning point of a parabola, or a node in a graph, depending on the context. |
| adjacency matrix | A square table recording edges between pairs of graph vertices. |
| adjacency list/əˈdʒeɪsənsi lɪst/ | A graph representation storing a list of neighbours for each vertex. |
| sparse | sparse |
| handshake lemma | In a finite undirected graph, the sum of vertex degrees equals twice the number of edges. |
| tree/triː/ | A connected undirected graph containing no cycles. |
| acyclic | acyclic |
| leaf | leaf |
| breadth-first search/bredθ fɜːst sɜːtʃ/ | breadth-first search |
| depth-first search/depθ fɜːst sɜːtʃ/ | depth-first search |
Friends, flights and build steps
- A social network, an airline map and the order in which code modules must compile look nothing alike. Each one is a set of things joined by links.
- That shared shape is a graph. Learn it once and the same tools count, search and check all three.
Vertices, edges and degree
- A graph 图 $G = (V, E)$ has vertices 顶点 $V$ and edges 边 $E$ joining pairs of them. Edges can be directed 有向 (one-way, like build order) or undirected (like friendship).
- The degree 度 of a vertex is the number of edges touching it. An isolated vertex has degree 0 and still counts as a vertex.
Match each system to the kind of graph that models it.
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Friendship runs both ways; build order runs one way and cannot loop; distances are weights; links can point back.
Two ways to store a graph
- An adjacency matrix 邻接矩阵 is an $n \times n$ grid of 0s and 1s. It takes $n^2$ space but answers "is there an edge from $u$ to $v$?" in $O(1)$.
- An adjacency list 邻接表 stores each vertex's neighbours. It takes $O(|V| + |E|)$ space, so it suits sparse 稀疏 graphs with few edges per vertex.
A road map has a million junctions, each joined to about four others. The best storage is —
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A matrix needs 10¹² cells, almost all 0; a list stores only the 4 million or so edges that exist.
The handshake lemma
- Each edge adds 1 to the degree of each of its two ends, so $\sum_v \deg(v) = 2|E|$. This is the handshake lemma 握手引理. In the figure, $3 + 2 + 2 + 2 + 3 = 12 = 2 \times 6$.
- Corollary: the number of odd-degree vertices is always even. Seven people cannot each shake hands with exactly three others, because $7 \times 3 = 21$ is odd.
A graph has 9 edges. What is the sum of all its vertex degrees?
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Each edge adds 2, so $2 \times 9 = 18$.
A graph can have exactly three vertices of odd degree.
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The degree sum is even, so the odd degrees must come in an even number.
Trees: connected, no cycles
- A tree 树 is connected (one piece) and acyclic 无环 (no cycles). Every tree with $|V|$ vertices has exactly $|E| = |V| - 1$ edges.
- A leaf 叶子 has degree 1. Combine the two facts with the handshake lemma to answer most tree questions.
A tree has 12 vertices, 5 of them leaves. Edges: $12 - 1 = 11$, so the degree sum is $2 \times 11 = 22$. The leaves give 5, so the other 7 vertices carry $22 - 5 = 17$.
A tree has 30 vertices. How many edges does it have?
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Every tree has $|E| = |V| - 1 = 29$.
Two ways to search
- Breadth-first search 广度优先搜索 (BFS) uses a queue and visits level by level. In an unweighted graph it finds shortest paths, counted in edges.
- Depth-first search 深度优先搜索 (DFS) uses a stack or recursion and goes as deep as it can before backing up. It finds connected pieces and detects cycles.
Which statements about BFS are true? Select all that apply.
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BFS counts edges, so it gives shortest paths only when every edge costs the same.
Marks that slip away
- Connected is not complete. Connected means one piece; complete means every pair is joined.
- BFS on weighted graphs. BFS counts edges, not weights; a weighted shortest path needs another method.
- Ignoring degree-0 vertices. They still count in $|V|$.
You've got it
- $G = (V, E)$; a matrix costs $n^2$ with $O(1)$ edge tests; a list costs $O(|V| + |E|)$, best for sparse graphs.
- $\sum \deg = 2|E|$, so the number of odd-degree vertices is even.
- Tree: connected and acyclic, $|E| = |V| - 1$. BFS: queue, level order; DFS: stack, depth first.