Okay, let's tackle this task. The user wants me to replace every word in the given text with three synonyms, using the syn3 format. But wait, the instructions say to keep brand names. In the provided text, there's a mention of $G_1$ and $G_2$. These are probably graph notations, not brand names, so they should stay as they are.
Therefore, the final answer should look like:
I need to ensure that replacements don't change the meaning, especially in the solution manual context. For example, "comprehensive solution manual" could become detailed solution guide. pearls in graph theory solution manual
Problem 3.1: "Prove that a graph is a tree if and only if it is connected and has no cycles." "Prove" => 显示, "connected" => 联动的, "has no cycles" => 不含环.
Next, terms like "connected" can be "连通的|连接的|联动的." "Cycle" could be "环|圈|循环." Okay, let's tackle this task
Wait, the problem statement includes a math problem: "Prove that a graph with $n$ vertices can have at most $\fracn(n-1)2$ edges." So equations should remain as is. Also, "Problem 1.1" is a label, so maybe leave the problem number untouched.
Looking at the mathematical terms like "graph" and "tree," they need appropriate synonyms. Maybe "图" can be "图例|网络|网络图." "Tree" could be "树|树形结构|木质结构," but I need to check if "tree" in graph theory has specific synonyms. Wait, in graph theory, "tree" is a specific term, so maybe the synonyms should be related terms instead of literal translations. Hmm, maybe "树" is already the standard term in Chinese, so perhaps there are no synonyms here. But the user might still want three options, so perhaps "树|树形结构|树状图." In the provided text, there's a mention of $G_1$ and $G_2$
I need to go through each part of the text and replace every term with three synonyms, keeping the mathematical notation intact. For example, "Suppose the graph is a tree. Then, it is connected by definition." becomes "假设网络图是树状图。则，它连通的根据定义。"