Distance Signatures of Extended and Co-extended Incidence Graphs of Affine Designs

The distance matrix of a connected graph [Formula: see text], denoted by [Formula: see text], is the matrix whose rows and columns are indexed by the vertex set [Formula: see text] such that the [Formula: see text]-entry is [Formula: see text], where [Formula: see text], [Formula: see text]. The distance signature [Formula: see text] of [Formula: see text] is the inertia of [Formula: see text]. In this paper, we determine the distance signature of the extended (co-extended) incidence graph of an affine design. Furthermore, we state that an open Graffiti conjecture is true for the extended (co-extended) incidence graphs of affine designs by investigating the lower bound of the matching number.