A family of symmetric graphs in relation to 2-point-transitive linear spaces

A graph $\mathrm{\Gamma }$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\mathrm{\Gamma }$, where an arc is an ordered pair of adjacent vertices. Let $\mathrm{\Gamma }$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition $\mathcal{B}$, and let $\mathcal{D}(\mathrm{\Gamma },\mathcal{B})$ be the incidence structure with point set $\mathcal{B}$ and blocks $\{B\}\cup {\mathrm{\Gamma }}_{\mathcal{B}}(\alpha )$, for $B\in \mathcal{B}$ and $\alpha \in B$, where ${\mathrm{\Gamma }}_{\mathcal{B}}(\alpha )$ is the set of blocks of $\mathcal{B}$ containing at least one neighbour of $\alpha$ in $\mathrm{\Gamma }$. In this paper we classify all $G$-symmetric graphs $\mathrm{\Gamma }$ such that ${\mathrm{\Gamma }}_{\mathcal{B}}(\alpha )\mathrm{\ne }{\mathrm{\Gamma }}_{\mathcal{B}}(\beta )$ for distinct $\alpha ,\beta \in B$, the quotient graph of $\mathrm{\Gamma }$ with respect to $\mathcal{B}$ is a complete graph, and $\mathcal{D}(\mathrm{\Gamma },\mathcal{B})$ is isomorphic to the complement of a $(G,2)$-point-transitive linear space.