  This paper introduces an extended notion of expansion suitable for radio
networks. A graph $G=(V,E)$ is called an $(\alpha_w, \beta_w)$-{wireless
expander} if for every subset $S \subseteq V$ s.t. $|S|\leq \alpha_w \cdot
|V|$, there exists a subset $S'\subseteq S$ s.t. there are at least $\beta_w
\cdot |S|$ vertices in $V\backslash S$ adjacent in $G$ to exactly one vertex in
$S'$. The main question we ask is the following: to what extent are ordinary
expanders also good {wireless} expanders? We answer this question in a nearly
tight manner. On the positive side, we show that any $(\alpha, \beta)$-expander
with maximum degree $\Delta$ and $\beta\geq 1/\Delta$ is also a $(\alpha_w,
\beta_w)$ wireless expander for $\beta_w = \Omega(\beta / \log (2 \cdot
\min\{\Delta / \beta, \Delta \cdot \beta\}))$. Thus the wireless expansion is
smaller than the ordinary expansion by at most a factor logarithmic in
$\min\{\Delta / \beta, \Delta \cdot \beta\}$, which depends on the graph
\emph{average degree} rather than maximum degree; e.g., for low arboricity
graphs, the wireless expansion matches the ordinary expansion up to a constant.
We complement this positive result by presenting an explicit construction of a
"bad" $(\alpha, \beta)$-expander for which the wireless expansion is $\beta_w =
O(\beta / \log (2 \cdot \min\{\Delta / \beta, \Delta \cdot \beta\})$.
  We also analyze the theoretical properties of wireless expanders and their
connection to unique neighbor expanders, and demonstrate their applicability:
Our results yield improved bounds for the {spokesmen election problem} that was
introduced in the seminal paper of Chlamtac and Weinstein (1991) to devise
efficient broadcasting for multihop radio networks. Our negative result yields
a significantly simpler proof than that from the seminal paper of Kushilevitz
and Mansour (1998) for a lower bound on the broadcast time in radio networks.
