PBIB-Designs and 2-Steiner Distance Eigenvalues of Split Graphs

In this paper, we have considered split graph \(K_{n}\circ K_{1}\) with exactly four distinct adjacency eigenvalues. Since these graphs are not regular, it is laborious to construct block designs out of them. In this paper, we have shown the existence (hence construction) and non-existence of block designs arising out of split graphs \(K_{n}\circ K_{1}\). In the second half of this paper, we have partially tried an \(NP\)-hard problem of finding 2-Steiner distance matrix of the split graph \(K_{n}\circ K_{1}\). Furthermore, we have given interlacing for adjacency, distance and 2-Steiner distance spectra of split graph \(K_{n}\circ K_{1}\). Time complexity and algorithm for finding rank of a matrix is given.