I have a undirected graph $(V,E)$, $V=\{1, \dots, d\}$, with set of maximal cliques $\mathcal C$. I am interested in the subspace of $\mathbb R^d$ spanned by the clique incidence vectors $x_C \in \{0,1\}^d$, $C \in \mathcal C$ where a 1 in place $j$ indicates clique membership of vertex $j$:
$$x_C = \sum_{v \in C} e_v$$
where $e_i$ is the $i$th unit vector.
For example for the cycle graph with 4 vertices $\{1,2,3,4\}$ the clique matrix $$\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 1 \\ 1 &0 & 0 & 1 \end{bmatrix}^T $$ of incidence vectors of cliques $(1, 1, 0, 0)$, $(0, 1, 1, 0)$, $(0, 0, 1, 1)$ and $(1, 0, 0, 1)$ has rank 3. What can I say about the dimension of the (sub-)space? Is there a condition that a.) the vectors are independent and b.) are full rank ($d$-dimensional space)?
