In this paper, we have studied the Signless Laplacian spectrum of the power graph of finite cyclic groups. We have shown that n À 2 is an eigen value of Signless Laplacian of the power graph of Z n , n ! 2 with multiplicity at least /ðnÞ: In particular, using the theory of Equitable Partitions, we have completely determined the Signless Laplacian spectrum of power graph of Z n for n ¼ pq where p, q are distinct primes.

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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].

The reduced power graph of a finite group [Formula: see text], denoted by [Formula: see text], is the graph whose vertices are the elements of the group [Formula: see text] and two vertices [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we first describe the structure of the reduced power graph of the finite cyclic group [Formula: see text]. Consequently, we provide a short and alternative proof of one of the results published in (R. Rajkumar and T. Anitha, Laplacian spectrum of reduced power graph of certain finite groups, Linear Multilinear Algebra (2019) 1–18). We characterize the values of [Formula: see text] for which [Formula: see text] is a line graph. We then deduce the signless Laplacian spectrum of [Formula: see text] using its structure. We provide lower and upper bounds on the signless Laplacian spectral radius of [Formula: see text]. Finally, we conclude the paper by determining the signless Laplacian spectrum of [Formula: see text], where [Formula: see text] denotes the dihedral group of order [Formula: see text].