Perfect octagon quadrangle systems with upper C4-systems

An octagon quadrangle is the graph consisting of an 8-cycle (x1, x2,…, x8) with two additional chords: the edges {x1, x4} and {x5, x8}. An octagon quadrangle system of order v   and index ρρ [OQS] is a pair (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks  ) which partition the edge set of ρKvρKv defined on X. An octagon quadrangle system  Σ=(X,H)Σ=(X,H) of order v   and index λλ is said to be upper C4-perfect if the collection of all of the upper  4-cycles contained in the octagon quadrangles form a μ-fold 4-cycleμ-fold 4-cycle system of order v; it is said to be upper strongly perfect, if the collection of all of the upper  4-cycles contained in the octagon quadrangles form a μ-foldμ-fold 4-cycle system of order v and also the collection of all of the outside  8-cycles contained in the octagon quadrangles form a ϱ-foldϱ-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems.