Dodecagon Quadrangle Systems having the most large spectrum

A dodecagon quadrangle is the graph consisting of two cycles: a 12-cycle (x1, x2,.., x12) and a 4-cycle (x1, x4, x7, x10). A dodecagon quadrangle system [DQS] of order v and index λ[DQSλ] is a pair Σ = (X; B), where X is a nite set of v vertices and B is a collection of edge disjoint dodecagon quadrangles (called blocks) which partitions the edge set of λKv, the complete multigraph with vertex set X. In [12] the authors determined the spectrum of DQSs having index λ = 1 and the spectrum of perfect DQSs in all the cases. In [14] we determine the spectrum of DQSs of index λ = 2n(2h + 1) > 1, for any h ∈ N and n = 0; 1; 2; 3. In this paper we complete the determination of the spectrum in all the remaining cases.