Counting Graphs

One day, Tanya was studying graph theory. She is very inquisitive, so the following problem soon came to her mind. Find the number of undirected unweighted connected simple graphs with $N$ vertices (numbered $1$ through $N$) and $M$ edges, such that for each $i$ ($2 \le i \le N$), the shortest path from vertex $1$ to vertex $i$ is unique and its length is equal to $A_i$. In other words, for eac