Distance Laplacian spectra of various graph operations and its application to graphs on algebraic structures

Abstract

In this paper, we determine the distance Laplacian spectra of graphs obtained by various graph operations. We obtain the distance Laplacian spectrum of the join of two graphs [Formula: see text] and [Formula: see text] in terms of adjacency spectra of [Formula: see text] and [Formula: see text]. Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is the union of two regular graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs [Formula: see text], where [Formula: see text], in terms of their adjacency spectra. As applications of the results obtained, we have determined the distance Laplacian spectra of some well-known classes of graphs, namely the zero divisor graph of [Formula: see text], the commuting and the non-commuting graph of certain finite groups like [Formula: see text] and [Formula: see text], and the power graph of various finite groups like [Formula: see text], [Formula: see text] and [Formula: see text]. We show that the zero divisor graph and the power graph of [Formula: see text] are distance Laplacian integral for some specific [Formula: see text]. Moreover, we show that the commuting and the non-commuting graph of [Formula: see text] and [Formula: see text] are distance Laplacian integral for all [Formula: see text].