I am reading this paper http://www.math.ucsd.edu/~fan/wp/cheeger.pdf by F.R.K Chung, on the third page, line (1), she talks about the quantity $$\frac{<h, \mathcal{L}h>}{<h,h>}$$, where $\mathcal{L}$ is the normalized Laplacian of a simple graph, and $h:V\rightarrow \mathbb{C}$ is a function from $V$ the set of vertices of $G$ to $ \mathbb{C}$.
I am assuming $< , >$ is some inner product. However, which inner product is this? Also, what exactly is $\mathcal{L}h$? Since $\mathcal{L}$ is a $|E|$ by $|V|$ matrix and $h$ is just a function from $V$ to $\mathbb{C}$, I don't see what $\mathcal{L}h$ is supposed to be. Any help is appreciated.
The normalized Laplacian matrix $\mathcal{L}(G)$ of a graph $G=(V,E)$ is a matrix of order $|V|\times|V|$ and NOT $|E|\times|V|$.