Europ. J. Combinatorics (1986) 7, 105-114 A Class of Graphs Containing the Polar Spaces A. BLOKHUIS, T. KLOKS AND H. WILBRINK We consider finite graphs with the property that there exists a constant e such that for every maximal clique M and vertex x not in M, x is adjacent to exactly e vertices in M. It is shown that these graphs have a highly geometric structure which in many ways resembles that of the polar spaces. 1. INTRODUCTION In this paper, all graphs occurring are finite, undirected, and without loops or multiple edges. In [ 6], F. Zara considers graphs satisfying the following two axioms: (Al) there is an integer m such that every maximal clique has cardinality m; (A2) there is an integer e such that for every maximal clique M and every vertex x eM, x is adjacent to precisely e vertices of M. In this paper, a Zara graph is a graph satisfying (Al) and (A2). The class of Zara graphs includes the collinearity graphs of the (finite) polar spaces, T(2m) and Lim) (here e = m- 2 in both cases), McLaughlin's graph on 275 vertices and many others (see [ 4] for a description of these graphs). For a complete list of all known reduced Zara graphs see [7] (a precise definition of the term 'reduced' will be given below). Inspection of this list shows that all these graphs are strongly regular. It will be shown here that necessarily every reduced Zara graph is strongly regular. In addition to this, it will be shown that Zara graphs have a highly geometric structure which is very reminiscent of the geometry of polar spaces. For example, Zara graphs have a Buekenhout diagram (c. f. [4]) The following three trivial observations are fundamental. (1) Let rt. T 2 , ••• , r. be Zara graphs (on disjoint vertex sets) with parameters (mi. eJ, i = 1, 2, ... , s, such that mi- ei does not depend on i. Then the direct sum s T=E9Ti i=l is also a Zara graph (with parameters m = m 1 + · · · + m. and e = e 1 + m 2 + · · + m.). Here, the direct sum graph r is the graph with vertex set i=l (graphs will be identified with their vertex sets), two vertices being adjacent in r if and only if they are in distinct ri or are adjacent in the same ri. (2) Let T be a Zara graph with parameters (m, e) and let t be some positive integer. Then a Zara graph tT with parameters ( tm, te) can be obtained by 'blowing up' all vertices with a factor t. More formally, take r x {1, ... , t} to be the vertex set of tT and call (x, i) and (y, j) adjacent if x = y and i >'= j, or if x and y are adjacent in r. (3) If r is a Zara graph with parameters (m, e) and C is a clique, then r(C) := {xE Fix- y, Vy E C} is a Zara graph with parameters ( m -I Cl, e -I Cl). 105 0195-6698/86/020105 + 10 $02.00/0 © 1986 Academic Press Inc. (London) Limited