On graphs satisfying a strong adacency property

ON GRAPHS SATISFYING A STRONG ADJACENCY PROPERTY Y. Ananchuen and L. Caccetta School of Mathematics and Statistics Curtin University of Technology GPO Box U1987 Perth 6001 Western Australia. Dedicated to the memory of Alan Rahilly, 1947 1992 Let m and n be nonnegative integers and k be a positive integer. A graph G is said to have property P*(m,n,k) if for any set of m + n distinct vertices of G there are exactly k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The case n = 0 is, of course, a generalization of the property in the Friendship Theorem. In this paper we show that, for m = n = 1, graphs with this property are the ( (k+t)\l so-called strongly regular graphs with parameters t ,k+t, t-l, t) for some positive integer t. In particular, we show the existence of such graphs. For m ~ 1, n ~ 1, and m + n ~ 3, we show that, there is no graph having property P*(m,n,k), for any positive integer k.