## Eigenvalues and Expansion of Regular Graphs

### Author: Nabil Kahale

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 $2\sqrt{k-1}$. For a linear-sized subset $X$ and integer $t\ge2$, we show that the set of nodes at distance $t$ from $X$ has size at least $k(k-2)(k-1)^{t-2}|X|/2$. Given a graph $H$, we exhibit a necessary and sufficient condition for the existence of a family of $k$-regular graphs having a given asymptotic bound on the second eigenvalue, and containing $H$ as an induced subgraph. As a byproduct, we obtain improved bounds on random walks on expanders, we construct selection networks (resp. extrovert graphs) of smaller size (resp. degree) than was previously known, and we construct highly expanding non-Ramanujan graphs.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1993/93-70.ps