An Introduction to the Distance Laplacian Spectra of Graphs

In this book, we determine the distance Laplacian spectra of graphs obtained by various graph operations.We first obtain the distance Laplacian spectrum of the join of two graphs in terms of adjacency spectra. Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is a union of two graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs. We then find the universal adjacency spectra of the join of two regular graphs with distinct degrees and the join of two regular graphs in which of the graphs is a union of two regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. We obtain the distance Laplacian spectrum of Indu-Bala product of graphs in terms of their Laplacian spectra. We then obtain the distance signless Laplacian spectrum of Indu-Bala product of graphs in terms of their signless Laplacian spectra. Finally, we prove that the distance energy of the complete split graph always increases when an edge is deleted from it.