Local 2-geodesic transitivity and clique graphs

Abstract

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 mK 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)] ∼ = mK 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).

Locally triangular graphs and rectagraphs with symmetry

“…First suppose that N π 0 is not 4-transitive. Then (N π 0 , n) is one of (PSL 2 (8), 9), (PΓL 2 (8), 9) or (PΓL 2 (32), 33) by [20], in which case K π F n 2 by Lemma 3.8 (ii). Since the heart of N π 0 over F 2 is irreducible by [25], it follows from Lemma 3.7 that Π Q n or n .…”

“…In this paper, we study and classify a family of locally rank 3 graphs associated with the class of rectagraphs (defined below). Note that locally disconnected locally rank 3 graphs were recently analysed in a more general setting by Devillers et al [9], but the graphs we are interested in are locally connected.…”

“…Let H S n where n 5. Then H is transitive of rank 3 on n 2 if and only if (H, n) is one of (S n , n) or (A n , n) for n 5, (PΓL 2 (8), 9), or (M n , n) for n = 11, 12, 23 or 24.…”

Finite 2-Geodesic Transitive Abelian Cayley Graphs

“…The study of 2-geodesic transitive graphs was initiated in [3], where the family of tetravalent connected 2-geodesic transitive graphs was classified. Later, in [4], the structure of [ (u)] (the induced subgraph on (u) of vertices adjacent to u) was determined for each 2-geodesic transitive graph .…”

“…We denote by K m [b] the complete multipartite graph with m parts, and each part has size b, where m ≥ 3, b ≥ 2, and K 3 [2] is the octahedron. (4) has been studied extensively, see [1,8,10,15,20].…”