陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「254B, Notes 1: Basic theory of expander graphs」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

The objective of this course is to present a number of recent constructions of expander graphs , which are a type of sparse but “pseudorandom” graph of importance in computer science, the theory of random walks, geometric group theory, and in number theory. The subject of expander graphs and their applications is an immense one, and 下面会 not possibly be able to cover it in full in this course. In particular, 下面会 say almost nothing about the important applications of expander g

Instead of focusing on applications, this course will concern itself much more with the task of constructing expander graphs. This is a surprisingly non-trivial problem. On one hand, an easy application of the probabilistic method shows that a randomly chosen (large, regular, bounded-degree) graph will be an expander graph with very high probability, so expander graphs are extremely abundant. On the other hand, in many applications, one wants an expander graph that is more de

已知结果和反例

(There are also other important constructions of expander graphs that are not related to Cayley or Schreier graphs, such as those graphs constructed by the zigzag product construction , but 下面会 not discuss those types of graphs in this course, again referring the reader to the survey of Hoory, Linial, and Wigderson .)

We begin by defining the concept of an expander graph formally. As with many fundamentally important concepts in mathematics, there are a number of equivalent definitions of this concept. We will adopt a “spectral” perspective towards expander graphs, defining them in terms of a certain spectral gap, but will relate this formulation of expansion to the more classical notion of edge expansion later in this section.

证明或构造的主线

We begin by recalling the notion of a graph. To avoid some very minor technical issues, 下面会 work with undirected, loop-free, multiplicity-free graphs (though later, when we discuss Cayley graphs, 下面会 allow loops and repetition).

Definition 1 A graph is a pair , where is a set (called the vertex set of ), and is a collection of unordered pairs of distinct elements of , known as the edge set of . Elements of or are called vertices and edges of A graph is finite if the vertex set (and hence the edge set) is finite. If is a natural number, we say that a graph is -regular if each vertex of is contained in exactly edges in ; we refer to as the degree of the regular graphs.

阅读时建议盯住的点

Example 2 The complete graph on a vertex set has edge set . If has elements, the complete graph is -regular.

In this course, 下面会 mostly be interested in constant-degree large finite regular graphs, in which is fixed (e.g. ), and the number of vertices is going off to infinity.

值得单独记下的条目

• Show that if and only if is not connected .
• Show that if and only if contains a non-empty bipartite graph as a connected component.
• (iii) Show that , where denotes a quantity that goes to zero as for fixed .
• Show that the eigenvalues of the adjacency operator associated to are for . ( Hint: you may find the discrete Fourier transform to be helpful.)
• Show that is not a one-sided expander family (and is thus not a two-sided expander family either). This is despite always being connected (and non-bipartite for odd).
• (i) form a one-sided expander family.
• (ii) There exists such that for all sufficiently large .
• (i) form a one-sided expander family.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源