## Abstract

The normalized distance Laplacian matrix of a connected graph G, denoted by D^{L} (G), is defined by D^{L} (G) = T r(G)^{−1/2} D^{L} (G)T r(G)^{−1/2}, where D(G) is the distance matrix, the D^{L} (G) is the distance Laplacian matrix and T r(G) is the diagonal matrix of vertex transmissions of G. The set of all eigenvalues of D^{L} (G) including their multiplicities is the normalized distance Laplacian spectrum or D^{L}-spectrum of G. In this paper, we find the D^{L}-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the D^{L}-spectrum of the graphs associated with algebraic structures. In particular, we find the D^{L}-spectrum of the power graphs of groups, the D^{L}-spectrum of the commuting graphs of non-abelian groups and the D^{L}-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

Original language | English |
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Pages (from-to) | 213-230 |

Number of pages | 18 |

Journal | Carpathian Journal of Mathematics |

Volume | 39 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2023 |

## Keywords

- Commuting graphs
- Joined union
- Normalized distance Laplacian matrix
- Power graphs
- Zero-divisor graphs

## ASJC Scopus subject areas

- General Mathematics