PBIB-Designs and 2-Steiner Distance Eigenvalues of Split Graphs - Huilgol, Medha Itagi; Asok, Sreepriya (Communications in Mathematics and Applications, 2024)
In this paper, we have considered split graph \(K_n° K_1\) 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 \(K_n° K_1\). 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 \(K_n° K_1\). Furthermore, we have given interlacing for adjacency, distance and 2-Steiner distance spectra of split graph \(K_n° K_1\). Time complexity and algorithm for finding rank of a matrix is given.