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Relative expanders or weakly relatively Ramanujan graphs. (English) Zbl 1035.05058

Let \(G\) be a fixed graph with largest eigenvalue \(\lambda_0\) and with universal cover having spectral radius \(\rho\). The author demonstrates that a random cover of large degree over \(G\) has its new eigenvalues bounded in absolute value by roughly \(\sqrt{\lambda_0\, \rho}\). The main result is a “relative version” of the Broder-Shamir bound on eigenvalues of random regular graphs.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C80 Random graphs (graph-theoretic aspects)
68R10 Graph theory (including graph drawing) in computer science
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