I'm trying to solve the following question:
Given that the bipartite graph $G=\left(L\cup R,E\right)$ is $d$-regular and satisfies $|L|=|R|=n$. Given that for every set of vertices $L'\subset L$ of size $|L'|\leq\frac{1}{2}n$ there are at least $|L'|$ Neighbors in $R$. Analyze $h_{v}\left(G\right)$.
I have the following definition:
Let $G=\left(V,E\right)$ be a graph with $n$ nodes and $1\leq k\le\frac{n}{2}$. The expansion ratio at nodes for sets of size up to $k$ is denoted by $h_{v}\left(G,k\right)$ and is defined as: $$ h_{v}\left(G,k\right)\triangleq\min_{S\subset V,0<|S|\leq k}\frac{|\Gamma\left(S\right)\backslash S|} {|S|}=\min_{S\subset V,0<|S|\leq k}\frac{\partial_{v}\left(S\right)}{|S|} $$ where $\Gamma\left(S\right)$ is the set of neighbors of $S$. In fact, we denote the orthographic language to be: $$ \partial_{v}\left(S\right)\triangleq\Gamma\left(S\right)\backslash S $$
But I can't seem to figure how to actually calculate $h_{v}\left(G\right)$ - I understand how the graph looks like in general - but how do I "translate" it into this forumla?