Merge 276e1413ca into 356cc35e8a

en/docs/chapter_graph/summary.md

@@ -1,14 +1,14 @@
 # Summary
 
-### Key review points
+### Key review
 
 - A graph is made up of vertices and edges. It can be described as a set of vertices and a set of edges.
 - Compared to linear relationships (like linked lists) and hierarchical relationships (like trees), network relationships (graphs) offer greater flexibility, making them more complex.
-- In a directed graph, edges have a direction. In a connected graph, any vertex can be reached from any other vertex. In a weighted graph, each edge has an associated weight variable.
+- In a directed graph, edges have directions. In a connected graph, any vertex can be reached from any other vertex. In a weighted graph, each edge has an associated weight variable.
-- An adjacency matrix is a way to represent a graph using matrix (2D array). The rows and columns represent the vertices. The matrix element value indicates whether there is an edge between two vertices, using $1$ for an edge or $0$ for no edge. Adjacency matrices are highly efficient for operations like adding, deleting, or checking edges. But they require more space.
+- An adjacency matrix is a way to represent a graph using matrix (2D array). The rows and columns represent the vertices. The matrix element value indicates whether there is an edge between two vertices, using $1$ for an edge or $0$ for no edge. Adjacency matrices are highly efficient for operations like adding, deleting, or checking edges, but they require more space.
 - An adjacency list is another common way to represent a graph using a collection of linked lists. Each vertex in the graph has a list that contains all its adjacent vertices. The $i^{th}$ list represents vertex $i$. Adjacency lists use less space compared to adjacency matrices. However, since it requires traversing the list to find edges, the time efficiency is lower.
-- When the linked lists in a adjacency list are long enough, they can be converted into red-black trees or hash tables to improve lookup efficiency.
+- When the linked lists in an adjacency list are long enough, they can be converted into red-black trees or hash tables to improve lookup efficiency.
-- From the perspective of algorithmic design, an adjacency matric reflects the concept of "trading space for time", whereas an adjacency list reflects "trading time for space".
+- From the perspective of algorithmic design, an adjacency matrix reflects the concept of "trading space for time", whereas an adjacency list reflects "trading time for space".
 - Graphs can be used to model various real-world systems, such as social networks, subway routes.
 - A tree is a special case of a graph, and tree traversal is also a special case of graph traversal.
 - Breadth-first traversal of a graph is a search method that expands layer by layer from near to far, typically using a queue.
@@ -28,4 +28,4 @@ In a disconnected graph, there is at least one vertex that cannot be reached fro
 
 **Q**: In an adjacency list, does the order of "all vertices connected to that vertex" matter?
 
-It can be in any order. However, in real-world applications, it might be necessary to sort them according to certain rules, such as the order in which vertices are added, or the order of vertex values. This can help quickly find vertices with certain extreme values.
+It can be in any order. However, in real-world applications, it might be necessary to sort them according to certain rules, such as the order in which vertices are added, or the order of vertex values. This can help find vertices quickly with certain extreme values.