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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}
