On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

Bilal A. Rather, Hilal A. Ganie, Mustapha Aouchiche

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

The normalized distance Laplacian matrix of a connected graph G, denoted by DL (G), is defined by DL (G) = T r(G)−1/2 DL (G)T r(G)−1/2, where D(G) is the distance matrix, the DL (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 DL (G) including their multiplicities is the normalized distance Laplacian spectrum or DL-spectrum of G. In this paper, we find the DL-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 DL-spectrum of the graphs associated with algebraic structures. In particular, we find the DL-spectrum of the power graphs of groups, the DL-spectrum of the commuting graphs of non-abelian groups and the DL-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

Original languageEnglish
Pages (from-to)213-230
Number of pages18
JournalCarpathian Journal of Mathematics
Volume39
Issue number1
DOIs
Publication statusPublished - 2023

Keywords

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

ASJC Scopus subject areas

  • General Mathematics

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