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

Abstract

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 [Formula: see text]-design, called a diametral-design (in short) over a strongly regular graph [Formula: see text] of degree [Formula: see text], diameter [Formula: see text], is an ordered pair [Formula: see text], where [Formula: see text] and [Formula: see text] is the set of all diametral paths of [Formula: see text], called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text] blocks containing [Formula: see text]; (ii) If [Formula: see text], [Formula: see text] and [Formula: see text], then there are exactly [Formula: see text]-blocks containing [Formula: see text]. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs [Formula: see text], the triangular graphs [Formula: see text], 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.