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Consider a directed graph $G$ and the associated Path-edge matrix $P_{i,j} \equiv \mathcal{X}(\text{edge $i$ contains path $j$})$, for $j\in \mathcal{P}$ (set of collection of all paths between pairs of vertices $\{(u_k,v_k)|k=1, \dots, K\}$) and $i\in E$ (the set of edges contained in $\mathcal{P}$). Here $\mathcal{X}$ represents the indicator function.

What is known about the rank of matrix P? Specifically, is it a full rank matrix for all connected graphs and all possible sets of vertex pairs?

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    Is $\mathcal{P}$ the set of all possible paths (all possible vertex ordered pairs), or just the set of all existing paths? – 2017-01-07
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    $\mathcal{P}$ is the set of all possible paths between the given $K$ vertex pairs, which may or may not be all the possible vertex pairs. I have an `intuition' that in some situations, while all possible pairs can make $P$ full rank, restricting ourselves to a specific set of vertex pairs can, potentially, make $P$ rank deficient. – 2017-01-07

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