Local 2-geodesic transitivity and clique graphs

A 2-geodesic in a graph is a vertex triple ( u , v , w ) such that v is adjacent to both u and w and u , w are not adjacent. We study non-complete graphs Γ in which, for each vertex u, all 2-geodesics with initial vertex u are equivalent under the subgroup of graph automorphisms fixing u. We call such graphs locally 2-geodesic transitive, and show that the subgraph [ Γ ( u ) ] induced on the set of vertices of Γ adjacent to u is either (i) a connected graph of diameter 2, or (ii) a union m K r of m ⩾ 2 copies of a complete graph K r with r ⩾ 1 . This suggests studying locally 2-geodesic transitive graphs according to the structure of the subgraphs [ Γ ( u ) ] . We investigate the family F ( m , r ) of connected graphs Γ such that [ Γ ( u ) ] ≅ m K r for each vertex u, and for fixed m ⩾ 2 , r ⩾ 1 . We show that each Γ ∈ F ( m , r ) is the point graph of a partial linear space S of order ( m , r + 1 ) which has no triangles (and 2-geodesic transitivity of Γ corresponds to natural strong symmetry properties of S ). Conversely, each S with these properties has point graph in F ( m , r ) , and a natural duality on partial linear spaces induces a bijection F ( m , r ) ↦ F ( r + 1 , m − 1 ) .