I have a feature space consisting of five amino acids in a protein structure. Three of five amino acids $(A, B, C)$ are within some cutoff distance of each other and two amino acids $(D, E)$ are spaced evenly apart. I consider each amino acid to be a node with closely spaced amino acids connected with paths. Therefore I have the adjacency matrix $A$:
\begin{equation} A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{equation}
I know that I can compute the Laplacian matrix $L$ from $A$. I can determine the number of disconnected components by determining the nullspace of $L$. The number of disconnected components is $3$ in this case.
My question is, how can I determine the number of paths in each disconnected component? Visually, I can see that the first disconnected component $(A, B, C)$ has two paths and both $(D)$ and $(E)$ have zero connecting paths. How can I determine this algebraically/computationally?