Constructing a Class of Symmetric Graphs

Abstract

We find a natural construction of a large class of symmetric graphs from point-and block-transitive 1-designs. The graphs in this class can be characterized as G-symmetric graphs whose vertex sets admit a G-invariant partition B of block size at least 3 such that, for any two blocks B, C of B, either there is no edge between B and C, or there exists only one vertex in B not adjacent to any vertex in C. The special case where the quotient graph B of relative to B is a complete graph occurs if and only if the 1-design needed in the construction is a G-doubly transitive and G-block-transitive 2-design, and in this case we give an explicit classification of when G is a doubly transitive projective group or an affine group containing the affine general group. Examples of such graphs include cross ratio graphs studied recently by Gardiner, Praeger and Zhou and some other graphs with vertices the (point, line)-flags of the projective or affine geometry.

Finite symmetric graphs with two‐arc transitive quotients II

“…Although B stores a lot of information about the original graph , a genuine picture of would need the bipartite subgraph induced on two adjacent blocks and a 1-design with point set B. ) mirrors the structure of , and this approach to imprimitive symmetric graphs was used in [16] and further developed in [17,24,20,32,33,35,36] The notation and terminology for graphs, groups, and designs used in the article are standard; see, for example, [4], [12], and [3], respectively. For a group G acting on a set and for X ⊆ , G X and G (X) are the setwise and pointwise stabilizers of X in G, respectively.…”

“…From [4, Chapter 19], a pair (K, φ) required in the construction above always exists. For each of (i)-(vi) above, all 3-arc graphs of the (H, 2)-arc transitive graph K b+1 have been determined in [35]. In the case where H is 4-transitive, we have either …”

“…See for example [7,8,14,15,[21][22][23]28]; [1,25,27] for 2-arc transitive Cayley graphs; and [17,20,24,33,35] for 2-arc transitive quotients of symmetric graphs. The present article is a contribution toward symmetric graphs with 2-arc transitive quotients, and it forms part of our study of imprimitive symmetric graphs.…”

Classification of a family of symmetric graphs with complete 2-arc-transitive quotients

“…This forms part of the broad program [9,11,12,[17][18][19][20] of studying symmetric graphs with 2-arc-transitive quotients. Of particular interest arising from the classification are two subfamilies of symmetric graphs which admit an arc-transitive action of a projective linear group.…”

Solution to a Question on a Family of Imprimitive Symmetric Graphs

“…In line with a geometrical approach suggested in [1], various situations may occur for , G, B , [B, C] and a certain 1-design with point set B; see, for example, [1,3,[5][6][7]. The case where k = v − 2 ≥ 1 was studied in [2,4] and a necessary and sufficient condition for B to be (G, 2)-arc-transitive was given in [2].…”