On the geometry of Zara graphs

Abstract

We show that, up to trivial examples, the definition of Zara graph (or a graph satisfying hypothesis (H) in F. Zara [European J. Combin. 5 (1984), 255–290]) can be weakened to the following axiom:

(Z) every maximal clique is regular in the sense of A. Neumaier [in “Finite Geometries and Designs” (P. J. Cameron, J. W. P. Hirschfeld, and D. R. Hugues, Eds.), pp. 244–259, Cambridge Univ. Press, Cambridge, U. K., 1981] (i.e., given a maximal clique, every vertex outside it is adjacent to the same number of vertices in it).