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

Subarsha Banerjee

Subarsha Banerjee

Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata 700019, India

E-mail Address: subarshabnrj@gmail.com

E-mail Address: subpm_rs@caluniv.ac.in

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https://doi.org/10.1142/S0219498823500226Cited by:2 (Source: Crossref)

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 $G_{1}$ $G_{1}$ and $G_{2}$ $G_{2}$ in terms of adjacency spectra of $G_{1}$ $G_{1}$ and $G_{2}$ $G_{2}$ . 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 $G_{i}$ $G_{i}$ , where $1 \leq i \leq n$ $1 \leq i \leq n$ , 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 $ℤ_{n}$ , the commuting and the non-commuting graph of certain finite groups like $D_{n}$ and ${Dic}_{n}$ , and the power graph of various finite groups like $ℤ_{n}$ , $D_{n}$ and ${Dic}_{n}$ . We show that the zero divisor graph and the power graph of $ℤ_{n}$ are distance Laplacian integral for some specific $n$ . Moreover, we show that the commuting and the non-commuting graph of $D_{n}$ and ${Dic}_{n}$ are distance Laplacian integral for all $n \geq 2$ .

Communicated by S. R. López-Permouth

Keywords:

AMSC: 05C25, 05C50

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