A characterization of partial geometric designs

AbstractA characterization of partial geometric designs with parameters (r, k, t, c) is given. It is shown that if N is the incidence matrix of a connected configuration (v, b, r, k), then the configuration is a partial geometric design (r, k, t, c), if and only if NN' has a single non-zero eigenvalue θ other than the simple eigenvalue rk. Then θ=r+k-1+c-t and its multipliciyt is α1= rk[(r-1)(k-1)-c] t(r+k-t-1-c). A necessary condition for the existence of a partial geometric design (r,k,t,c) is that α1 is a positive integer