We give new examples of graphs with the n-e.c. adjacency property. Few explicit families of n-e.c. graphs are known, despite the fact that almost all finite graphs are n-e.c. Our examples are collinearity graphs of certain partial planes derived from affine planes of even order. We use probabilistic and geometric techniques to construct new examples of n-e.c. graphs from partial planes for all n, and we use geometric techniques to give infinitely many new explicit examples if n = 3. We give a new construction, using switching, of an exponential number of non-isomorphic n-e.c. graphs for certain orders.

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w 1. IntroductionLet (5 be a coltineation group of a finite projective plane rc=(~, Y, I). By orbits, point orbits, line orbits and point-line orbits we shall mean N-orbits, N-orbits in ~,ffi-orbits in A '~, and ffi-orbits in N x ~ respectively. By homologies and elations we shNl mean homotoNes and elations in lb.In [6, pp. 183-t84], Fred Piper has proved that ifN fixes no point or line and if(~ contains order p elations for p>2, then the centers of order p elations form a point orbit, the axes of order p elations form a line orbit, and the centeraxis pairs of order p elations form a point-line orbit. Our main resutt, Theorem 6, is the exact analogue of Piper's result for order q homologies provided that q>2 and (~ fixes no point or line. The restriction that 15 fix no point or line is necessary. Otherwise more orbits can occur and in Theorem 4 we easily prove that at most three orbits (of each type) can occur if q> 2. The restriction that q > 2 is also necessary in Theorem 6. The main argument in the proof of Theorem 5 gives a partial result (which is included in the statement of Theorem 5), in the case q = 2. In a later paper [3] (or see Chapter III, Section 6, pp, 95-103 of [2]), we shall use this partial result together with some further arguments to prove that if q = 2 and (5 fixes no point or line then there are at most two orbits of centers, axes and center-axis pairs of order 2 homologies, The two orbit case does occur, examples are described in Mitchell [4,.I wish to thank Professor Andrew Gleason for his helpful suggestions while t was doing this research. w 2. Definitions, Notations, and Results Used ~z = (~, 5g, I) is a finite projective plane defined by the usual three axioms. Points are elements of ~ and will be denoted by capital letters P, Q .... Lines are elements of cS and will be denoted by small letters k, m ..... The incidence relation I wilt be used symmetrically, i.e., both PIk, kip wilt be used for "P and k are incident". The usual geometric language will also be used for incidence. AB means the unique line incident with the distinct points A and B. k m means the unique point incident with the distinct lines k and m. (k) is the set of points incident with k. (P) is the set of lines incident with P.