Let $\Gamma = (P,L,I)$ be a point-line geometry (here, $P$ is the point set, $L$ the line set, and $I$ is the symmetric incidence relation). (As an example, $\Gamma$ could be a graph.) I suppose $\vert P \vert = \vert L \vert = n \in \mathbb{N}$. Now put $P = \{ p_1,\ldots,p_n \}$ and $L = \{ l_1,\ldots,l_n\}$; according to this labeling, I define the incidence matrix of $\Gamma$ as the $(n \times n)$-matrix $A = (a_{ij})$ in which $a_{ij} = 1$ if $p_i\ I\ l_j$, and $0$ otherwise. I have the following two questions:
- Question 1: When is the determinant of $A$ (over $\mathbb{Q}$) equal to $1$ or $-1$ ? (So how does this translate to properties of the geometry $\Gamma$ ?)
- Question 2: Define the automorphism group $Aut(\Gamma)$ in the usual way. How can we see on $A$ that this group is transitive on lines and/or points ?
THANKS !
