A class of graphs containing the polar spaces (Q1084420)

A Zara graph is a finite undirected graph with the properties that there is an integer m such that every maximal clique has the cardinality m and that there is an integer e such that for every maximal clique M and every vertex \(x\not\in M\) the vertex x is adjacent to precisely e vertices of M. A certain closure operator \(S\mapsto S^{\perp \perp}\) on the set of subsets of the vertex set of a graph is introduced. If C is the vertex set of a clique, then so is \(C^{\perp \perp}\) and it is called a singular subspace. Some related concepts are introduced. The paper studies various properties of Zara graphs with respect to the introduced concepts. The results are divided into three items: The combinatorics of Zara graphs, The geometry of Zara graphs, Regularity properties of Zara graphs.