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\hyperbaseurl{http://toc15.dsteurer.org/}
\notag
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Homework 4 (pdf version)
Spectral gap under addition
In the exercise, you are to show that adding an expander to an arbitrary graph yields an expander.
Let \(G_1\) and \(G_2\) be \(d\)-regular graphs with the same set of vertices \(V\). Construct a \(2d\)-regular graph \(H\) by adding the edges of \(G_2\) to \(G_1\). Note that the normalized adjacency matrix of \(H\) satisfies \(A_H = (A_{G_1} + A_{G_2})/2\).
Suppose \(G_2\) satisfies \(\lambda(G_2)\le 1-\e\) for some \(\e>0\). Show that then \(H\) satisfies \(\lambda(H)\le 1-\e/2\).
Combinatorial characterization of positive spectral gap
Show that a regular graph \(G\) is connected and non-bipartite if and only if \(\lambda(G)<1\).
Combinatorial expansion of random regular graphs
Do Exercise 21.11 in the textbook.