New geometries for finite groups and polytopes

Isometric Point-Circle Configurations on Surfaces from Uniform Maps

“…The points and edges of this polygon are the points and lines of the generalized quadrangle with parameters (2, 1) (a grid on 9 points with 3 points on each line and 2 lines through each point). In [18] neighborhood geometries were defined for geometries as well as for graphs. Note that the neighborhood geometry of the generalized quadrangle (as a geometry) equals the neighborhood geometry of its symmetric, distance-regular collinearity graph obtained by forgetting the geometric structure.…”

“…Consider the (rank two) set system of points and blocks where the points are the vertices and the blocks are the neighborhoods of the vertices of an r-regular graph on v vertices. Such set systems are called neighborhood geometries of graphs, and were first defined in [18], within a more general context. If all neighborhoods are distinct, then the system has the following two properties: 1) each vertex appears in r blocks and 2) each block contains r vertices.…”

“…All the properties described above can be found in [18], although it is there erroneously stated that these geometries are always self-dual (it is true for example when the graph is not bipartite and when the automorphism group of the graph is arc-transitive).…”

On the rank two geometries of the groups PSL(2, q): part I

“…Hence, this geometry is clearly a complete graph and its Buekenhout diagram follows. Applying Corollary 4.1 of [19] to Γ 2 , we get Γ 1 and the corresponding Buekenhout diagram (see [19], Table 1 or [21]). The (IP ) 2 condition is clearly satisfied in Γ 2 and not in Γ 1 .…”

Invariant adjacency matrices of configuration graphs

“…where In, Jn, and A are the identity matrix, the all one matrix, and a (0, 1)-matrix with all row and column sums equal to κ, respectively. If A is an incidence matrix of some configuration C of type nκ, then the left-hand side Θ(A) := (κ−1)In +Jn −AA T is an adjacency matrix of the non-collinearity graph Γ of C. In certain situations, Θ(A) is also an incidence matrix of some nκ configuration, namely the neighbourhood geometry of Γ introduced by Lefèvre-Percsy, Percsy, and Leemans [9].…”