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\documentclass{acmabs}
\begin{document}
\Journal{Journal of the ACM}
\refkey{Kahale95}
\title{Eigenvalues and Expansion of Regular Graphs}
\author{Nabil Kahale}
\Month{September}
\Year{1995}
\Volume{42}
\Number{5}
\note{To appear}
\maketitle
\begin{abstract}
The spectral method is the best currently known technique to prove lower bounds
on expansion. Ramanujan graphs, which have asymptotically optimal second
eigenvalue, are the best known explicit expanders. The spectral method
yielded a lower bound of $k/4$ on the expansion of linear sized subsets of
$k$-regular Ramanujan graphs. We improve the lower bound on the expansion of
Ramanujan graphs to approximately $k/2$. Moreover, we construct a family of
$k$-regular graphs with asymptotically optimal second eigenvalue and linear
expansion equal to $k/2$. This shows that $k/2$ is the best bound one can
obtain using the second eigenvalue method. We also show an upper bound of
roughly $1 + \sqrt{k-1}$ on the average degree of linear-sized induced
subgraphs of Ramanujan graphs. This compares positively with the classical
bound $2\sqrt{k-1}$. As a byproduct, we obtain improved results on random
walks on expanders and construct selection networks (respectively extrovert
graphs) of smaller size (respectively degree) than was previously known.
\end{abstract}
\begin{categories}
F.2.2 [\textbf{Analysis of Algorithms and Problem Complexity}]:
Nonnumerical Algorithms and Problems; G.2.2 [\textbf{Discrete
Mathematics}]: Graph Theory
\end{categories}
\begin{terms}
Algorithms, Theory
\end{terms}
\begin{keywords}
Eigenvalues, expander graphs, induced subgraphs, load balancing, Ramanujan
graphs, random walks, selection networks
\end{keywords}
\end{document}