In a paper here (page $6$, lemma $2$) I am reading, there is a graph defined for a Hermitian matrix $H$ as ....
Consider a Hermitian matrix $H$ of dimension $2^n \times 2^n$ for some positive integer $n$, where there are no more than $d$ nonzero entries in each column. Let $G_H$ be the graph associated with $H$ whose vertex set is indexed by $\{0,1\}^n$. Each vertex corresponds to a row or column number, and there is an edge between vertex $x$ and $y$ if the matrix element $H_{x,y}$ is non-zero, and since $H$ is Hermitian, then necessarily $H_{y,x}$ is non-zero.
The author then states : "edge labels are then indexed from the set $\{1, \ldots, d \}^2$". This is where I am confused. For a given column, there will be at most $d$ entries, so that if $y$ is a vertex representing a column value, then it will have $d$ neighbours. But if $y$ represents a row value, then a given row can have more than $d$ entries, so I am not sure what is going on here. Moreover there are $2^n$ columns and $2^n$ rows. I would think there would since there is at most $2^n d$ entries, then the edges would be labelled $(\{0,1\}^n \times \{0,1, \cdots d\})$
Afterwards the author writes : "If $(x,y)$ is an edge of $G_H$ such that $y$ is the $i$'th neighbor of $x$, and $x$ is the $j'$th neighbor of $y$, then the edge $(x,y)$ is labeled with the ordered pair $(i,j)$ for $x \leq y$, or $(j,i)$ for $x \geq y$".
Any insights appreciated.