Undirected Graphs （无向图）
- G = (V, E)
- V = nodes.
- E = edges between pairs of nodes.
- Captures pairwise relationship between objects. （捕获对象之间的成对关系）
- Graph size parameters: n = |V|, m = |E|. （图的大小参数:n = |V|， m = |E|）
Some Graph Applications
Graph Representation: Adjacency Matrix (邻接矩阵)
Adjacency matrix. n-by-n matrix with A_uv= 1 if (u, v) is an edge. (当(u, v)是边时，n×n矩阵A_uv= 1)
- Two representations of each edge. (每条边有两种表示)
- Space proportional to n^2. （空间与n^2成比例）
- Checking if (u, v) is an edge takes Θ(1) time. （检查(u, v)是否为边需要Θ(1)时间）
- Identifying all edges takes Θ(n^2) time. （识别所有边需要Θ(n^2)时间）
Graph Representation: Adjacency List (邻接表)
Adjacency list. Node indexed array of lists. (链表的节点索引数组)
- Two representations of each edge. (每条边有两种表示)
- Space proportional to m + n. （空间与m + n成比例）
- Checking if (u, v) is an edge takes O(deg(u)) time. (degree = number of neighbors of u) // 检查(u, v)是否为边需要O(deg(u))时间。(deg = u的邻居数量)
- Identifying all edges takes Θ(m + n) time. (识别所有边需要Θ(m + n)时间)
Paths and Connectivity
Def. A path in an undirected graph G = (V, E) is a sequence P of nodes v_1, v_2, …, v_k-1, v_k with the property that each consecutive pair v_i, v_i+1 is joined by an edge in E.
(定义：无向图G = (V, E)中的一条路径是节点v_1, v_2，…，v_k-1, v_k的序列P，其性质是每一对连续的v_i, v_i+1在E中由一条边连接)
Def. A path is simple if all nodes are distinct.
(定义：如果所有节点都是独特的，则称路径为 简单的）
Def. An undirected graph is connected if for every pair of nodes u and v, there is a path between u and v.
(定义：如果对于每一对结点u和v，在u和v之间存在一条路径，则无向图是连通的)
Cycles
Def. A cycle is a path v_1, v_2, …, v_k-1, v_k in which v_1= v_k, k > 2, and the first k-1 nodes are all distinct.
(定义：一个循环是一个路径v_1, v_2，…，v_k-1, v_k，其中v_1= v_k, k > 2，并且第一个k-1节点都是不同的)
Trees
Def. An undirected graph is a tree if it is connected and does not contain a cycle.
(定义: 无向图是树，当它是连通的并且不包含循环)
Theorem. Let G be an undirected graph on n nodes. Any two of the following statements imply the third.
(定理: 设G是n个节点上的无向图。则下列任何两个陈述都可推出第三个)
- G is connected.
- G does not contain a cycle.
- G has n-1 edges.
Rooted Trees
Rooted tree. Given a tree T, choose a root node r and orient each edge away from r.
(根树: 给定一个树T，选择一个根节点r，并使每条边远离r)
Importance. Models hierarchical structure.
(重要性: 根树是一种分层次的结构)
