Partially balanced incomplete block (PBIB)-designs arising from diametral paths in some strongly regular graphs.

The partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well-studied concept. In this paper, we define and construct PBIB-designs with association schemes in strongly regular graphs through the nodes belonging to the diametral paths as blocks. A (v , b , r , k , λ , μ) -design, called a diametral-design (in short) over a strongly regular graph G = (V , E) of degree d , diameter diam (G) , is an ordered pair D = (V , B) , where V = V (G) and B is the set of all diametral paths of G , called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If x , y ∈ V and { x , y } ∈ E , then there are exactly λ blocks containing { x , y } ; (ii) If x , y ∈ V , x ≠ y and { x , y } ∉ E , then there are exactly μ -blocks containing { x , y }. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs L 2 (n) , the triangular graphs T (n) , the three Chang graphs, the Petersen graph, the Shrikhande graph, the Clebsch graph, the complement of Clebsch graph, the complement of Schläfli graph, complete bipartite graphs, the Hoffman–Singleton graph, etc. and in each case we give the composition of the diametral paths. These serve as extensions for the collection of near impossible class of strongly regular graphs with given parameters having two-association schemes. [ABSTRACT FROM AUTHOR]