Designs associated with maximum independent sets of a graph

A (v, βo, μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, βo, μ)-designs over the graphs Kn × Kn, T(n)-triangular graphs, L2(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, βo, μ)-designs over the three Chang graphs T1(8), T2(8) and T3(8).