z_0, z_1, \cdots, z_d = y \in X, 
z_{j-1} \textrm{\ and\ } z_j \textrm{\ are\ connected}\}.\end{aligned}$$ For a subset $Y \subseteq X$, $N_1(Y)$ is the collection of all the vertices which are connected with $Y$ by edges. A sequence $\{(X_m, E_m)\}_{m \in \mathbb{N}}$ of finite graphs is called an expander sequence, if the following conditions hold:

1.  The graphs $(X_m, E_m)$ are connected,

2.  There exists a natural number $D$ such that the graphs $(X_m, E_m)$ are all $D$-regular. Namely, for every $m$ and $x \in X_m$, $\sharp(\partial_1(x)) = D$,

3.  \[DefinitionExpanderCardinality\] $\lim_m \sharp (X_m) = \infty$,

4.  \[DefinitionExpanderExpansion\] There exists a constant $c > 0$ such that for every subset $Y \subseteq X_m$ satisfying $\sharp(Y) \le \sharp(X_m) / 2$, the inequality $\sharp(N_1(Y))  > (1 + c) \sharp(Y)$ holds.

Fix distance $S > 0$. For a subset $Y \subseteq X_m$, if $\mathrm{diam} (Y) < S$ and if $m$ is large enough, then $\sharp(Y) \le \sharp (X_m) / 2$ by condition (\[DefinitionExpanderCardinality\]). By condition (\[DefinitionExpanderExpansion\]), an expander sequence is a sequence of weak expander spaces.

A sequence of expander graphs is characterized in two ways. In the above description, we used a kind of negation of the Følner condition. A sequence 