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

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