Centrally symmetric Hasse diagrams of finite modular lattices
Bohdan Zelinka · Czechoslovak Mathematical Journal · 1970
In [3] a centrally symmetric graph, or 5-graph, is defined as an undirected graph without loops and multiple edges fulfilling the following conditions: (1) G contains at least one edge; (2) for each triplet {x, y, z} of its verticex such that дс{у ^ z) = 1 we have QG{X, У) 4= (3) for each vertex x of G exactly one vertes x exists such that for each vertex w of a neighbourhood of 3c we have QG{X, ̂ )> Qoi^ ^ ^)-Here QG{^, b) denotes the distance of a and b in G. The vertices x and 3c are called opposite to each other. In [3] the following theorems are proved. (A) If for each chosen vertex x of G there exists a Jordan-Dedekind lattice such that its Hasse diagram (see [2], [4]) is isomorphic to G and its greatest element is x, then G is an S-graph. (B) / / G is an arbitrary S-graph and x is its vertex, then x = x. (C) If G is an arbitrary S-graph and d is its diameter, then arbitrary two opposite vertices and only such two vertices have the distance d. Further in [3] A. KOTZIG suggests to study such S-graphs which satisfy the assump