A849. Counting Graphs--Platinum

Bessie has a connected, undirected graph $G$ with $N$ vertices labeled
$1\ldots N$ and $M$ edges ($2\le N\le 10^2, N-1\le M\le \frac{N^2+N}{2}$). $G$
may contain self-loops (edges from nodes back to themselves), but no parallel
edges (multiple edges connecting the same endpoints).
Let $f_G(a,b)$ be a boolean function that evaluates to true if there exists a
path from vertex $1$ to vertex $a$ that traverses exactly $b$ edges for each
$1\le a\le N$ and $0\le b$, and false otherwise. If an edge is traversed
multiple times, it is included that many times in the count.
Elsie wants to copy Bessie. In particular, she wants to construct an
undirected graph $G'$ such that $f_{G'}(a,b)=f_G(a,b)$ for all $a$ and $b$.
Your job is to count the number of distinct graphs $G'$ that Elsie may create,
modulo $10^9+7$. As with $G$, $G'$ may contain self-loops but no parallel
edges (meaning that there are $2^{\frac{N^2+N}{2}}$ distinct graphs on $N$
labeled vertices in total).
Each input contains $T$ ($1\le T\le \frac{10^5}{4}$) test cases that should be
solved independently. It is guaranteed that the sum of $N^2$ over all test
cases does not exceed $10^5$.

The first line of the input contains $T$, the number of test cases.
The first line of each test case contains the integers $N$ and $M$.
The next $M$ lines of each test case each contain two integers $x$ and $y$
($1\le x\le y\le N$), denoting that there exists an edge between $x$ and $y$
in $G$.
Consecutive test cases are separated by newlines for readability.

For each test case, the number of distinct $G'$ modulo $10^9+7$ on a new line.

In the first test case, $G'$ could equal $G$ or one of the two following
graphs:
5 4
1 2
1 4
3 4
3 5
5 5
1 2
2 3
1 4
3 4
3 5