On Laplacian integrability of comaximal graphs of commutative rings

Bilal Ahmad Rather, Mustapha Aouchiche, Muhammed Imran

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


For a commutative ring R, the comaximal graph Γ(R) of R is a simple graph with vertex set R and two distinct vertices u and v of Γ(R) are adjacent if and only if aR+bR=R. In this article, we find the Laplacian eigenvalues of Γ(Zn) and show that the algebraic connectivity of Γ(Zn) is always an even integer and equals ϕ(n), thereby giving a large family of graphs with integral algebraic connectivity. Further, we prove that the second largest Laplacian eigenvalue of Γ(Zn) is an integer if and only if n=pαqβ, and hence Γ(Zn) is Laplacian integral if and only if n=pαqβ, where p, q are primes and α,β are non-negative integers. This answers a problem posed by [Banerjee, Laplacian spectra of comaximal graphs of the ring Zn, Special Matrices, (2022)].

Original languageEnglish
Pages (from-to)310-324
Number of pages15
JournalIndian Journal of Pure and Applied Mathematics
Issue number1
Publication statusPublished - Mar 2024


  • 05C25
  • 05C50
  • 15A18
  • Algebraic connectivity
  • Comaximal graphs
  • Euler’s totient function
  • Integers modulo ring
  • Laplacian integral graphs
  • Laplacian matrix

ASJC Scopus subject areas

  • Applied Mathematics
  • General Mathematics


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