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\hyperbaseurl{http://toc15.dsteurer.org/}
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We will describe the zig-zag product of graphs, which allows us to reduce the degree of a graph while approximately maintaining its eigenvalue gap.
Let \(G\) be a regular graph with \(n\) vertices and degree \(D\). (Think of \(n\) as a growing parameter and \(D\) as large but absolut constant.)
Let \(H\) be a regular graph with \(D\) vertices and degree \(d\). (Think of \(d\) as a relatively small absolut constant, e.g., \(d=100\).)
Suppose that \(H\) is a very good expander, i.e., eigenvalue gap close to \(1\). (Since \(H\) has only constant size and we know that graphs with these properties exist, we could compute \(H\) efficiently by brute force.)
We will consider graphs with vertex set \([n]\times [D]\). We think of this set as \(n\) disjoint clouds of size \(D\).
The first graph we consider is \(I_n\otimes H\), which consists of \(n\) disjoint copies of \(H\), one copy per cloud. (The notation \(I_n\otimes H\) stems from the fact that the random walk matrix of the graph is the tensor product of the matrix \(I_n\) and the random walk matrix of \(H\).)
Next, we consider a graph \(\hat G\) obtained from \(G\) by splitting every vertex into \(D\) new vertices, one for each edge. In other words, \(\hat G\) is a perfect matching on \([n]\times [D]\) such that contracting each cloud yields the graph \(G\).
The idea of the zig-zag product is to combine the two graphs \(\hat G\) and \(I_n \otimes H\) to obtain a graph on \([n]\times [D]\) with much smaller degree than \(G\) but approximately the same eigenvalue gap.
Why should the graphs \(\hat G\) and \(I_n\otimes H\) be helpful? One good thing is that both graphs have small degrees. In fact, \(\hat G\) has only degree \(1\). Another good thing is that from far away (i.e., if we contract the clouds), the graph \(\hat G\) looks exactly like \(G\) (recall that we wanted the new graph to have the same eigenvalue gap as \(G\)). In the zig-zag construction, the graph \(I_n\ot H\) allows us to effectively contract the clouds, while maintaining small degree.
Definition: The zig-zag product of \(G\) with \(H\) is the graph \[G\boxtimes H=(I_n\ot H)\cdot \hat G \cdot (I_n\ot H).\]
The following lemma shows that \(G\boxtimes H\) and \(G\) have the same eigenvalue gap up to a \(\gamma(H)^2\) factor. (If we choose \(H\) carefully, then \(\gamma(H)\approx 1\).)
Lemma: \(\gamma(G\boxtimes H)\ge \gamma(G)\cdot \gamma(H)^2\).
Proof:
We will use the following useful characterization of the eigenvalue gap: A graph has eigenvalue gap at least \(\gamma\) if and only if its random walk matrix is a convex combination of the walk matrix of the complete graph and a matrix with largest eigenvalue at most \(1\) such that the walk matrix of the complete graph has weight at least \(\gamma\).
Hence, we can write \(H=\gamma_H J_D +(1-\gamma_H)E_H\) for \(\gamma_H=\gamma(H)\) and a matrix \(E_H\) with largest eigenvalue at most \(1\).
Using this decomposition for \(H\), we can see that \(G\boxtimes H\) is a convex combination of four matrices, \[ G \boxtimes H = \gamma_H^2 (I_n \ot J_D) \hat G (I_n \ot J_D)\] \[ \quad +~ \gamma_H(1-\gamma_H)(I_n \ot J_D) \hat G (I_n \ot E_H)\] \[ \quad +~ (1-\gamma_H)\gamma_H (I_n \ot E_H) \hat G (I_n \ot J_D)\] \[ \quad +~(1-\gamma_H)^2(I_n \ot E_H) \hat G (I_n \ot E_H).\] All four matrices have eigenvalues at most \(1\). Hence, \[ G \boxtimes H = \gamma_H^2 (I_n \ot J_D) \hat G (I_n \ot J_D) + (1-\gamma_H^2) E\] for a matrix \(E\) with eigenvalues at most \(1\).
How does the graph \((I_n \ot J_D) \hat G (I_n \ot J_D)\) look like? We claim that this graph is essentially \(G\). (The reason is that the multiplications with \((I_n\ot J_D\) effectively contract the clouds and we already noted that this contraction makes \(\hat G\) into \(G\).) Formally, \((I_n \ot J_D) \hat G (I_n \ot J_D)=G\ot J_D\).
The graph \(G\ot J_D\) has eigenvalue gap \(\gamma_G=\gamma(G)\). Hence, we can write it as a convex combination \(G\ot J_D= \gamma_G \cdot J_{Dn} + (1-\gamma_G)E'\) for a matrix \(E'\) with eigenvalues at most \(1\).
It follows that \(G\boxtimes H\) is a convex combination of the three matrices \(J_{D n}\), \(E\), and \(E'\) (all with eigenvalues at most \(1\)). The matrix \(J_{Dn}\) has weight \(\gamma_H^2\cdot \gamma_G\) in this convex combination. Thus, \(G\boxtimes H\) has eigenvalue gap at least \(\gamma_H^2\cdot \gamma_G\).