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en/docs/chapter_graph/summary.md

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 ### 1.   Key Review
 
 - Graphs consist of vertices and edges and can be represented as a set of vertices and a set of edges.
-- Compared to linear relationships (linked lists) and divide-and-conquer relationships (trees), network relationships (graphs) have a higher degree of freedom and are therefore more complex.
-- Directed graphs have edges with directionality, connected graphs have all vertices reachable from any vertex, and weighted graphs have edges that each contain a weight variable.
+- Compared with the linear relationships modeled by linked lists and the divide-and-conquer relationships modeled by trees, the network relationships modeled by graphs offer much greater flexibility and are therefore more complex.
+- In directed graphs, edges have direction; in connected graphs, every vertex is reachable from any other vertex; and in weighted graphs, each edge carries a weight.
 - Adjacency matrices use matrices to represent graphs, where each row (column) represents a vertex, and matrix elements represent edges, using $1$ or $0$ to indicate whether two vertices have an edge or not. Adjacency matrices are highly efficient for addition, deletion, lookup, and modification operations, but consume significant space.
-- Adjacency lists use multiple linked lists to represent graphs, where the $i$-th linked list corresponds to vertex $i$ and stores all adjacent vertices of that vertex. Adjacency lists are more space-efficient than adjacency matrices, but have lower time efficiency because they require traversing linked lists to find edges.
+- Adjacency lists use multiple linked lists to represent a graph: the $i$-th linked list corresponds to vertex $i$ and stores all vertices adjacent to it. Compared with adjacency matrices, adjacency lists use less space, but edge lookups are less efficient because the linked list must be traversed.
 - When linked lists in adjacency lists become too long, they can be converted to red-black trees or hash tables, thereby improving lookup efficiency.
 - From an algorithmic perspective, adjacency matrices embody "trading space for time", while adjacency lists embody "trading time for space".
 - Graphs can be used to model various real-world systems, such as social networks and subway lines.
 - Trees are a special case of graphs, and tree traversal is a special case of graph traversal.
-- Breadth-first search of graphs is a near-to-far, layer-by-layer expansion search method, typically implemented using a queue.
-- Depth-first search of graphs is a search method that prioritizes going as far as possible and backtracks when no path remains, commonly implemented using recursion.
+- Breadth-first search in graphs explores from near to far, expanding layer by layer, and is typically implemented with a queue.
+- Depth-first search in graphs follows a path as deep as possible and backtracks when it can go no farther, and is commonly implemented with recursion.
 
 ### 2.   Q & A
 
@@ -28,8 +28,8 @@ In this text, a path is viewed as a sequence of edges, not a sequence of vertice
 
 **Q**: In a disconnected graph, will there be unreachable vertices?
 
-In a disconnected graph, starting from a certain vertex, at least one vertex cannot be reached. Traversing a disconnected graph requires setting multiple starting points to traverse all connected components of the graph.
+In a disconnected graph, if you start from one vertex, at least one other vertex will be unreachable. To traverse a disconnected graph, you need multiple starting points so that all connected components are covered.
 
-**Q**: In an adjacency list, is there a requirement for the order of "all vertices connected to that vertex"?
+**Q**: In an adjacency list, is there any required ordering for the vertices adjacent to a given vertex?
 
-It can be in any order. However, in practical applications, it may be necessary to sort according to specified rules, such as the order in which vertices were added, or the order of vertex values, which helps quickly find vertices "with certain extreme values".
+They can appear in any order. In practice, however, they may need to be sorted according to specific rules, such as the order in which vertices were added or the order of vertex values, which helps when quickly finding a vertex with some extreme value.