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
JournalIndian Journal of Pure and Applied Mathematics
Publication statusAccepted/In press - 2023


  • Algebraic connectivity
  • Comaximal graphs
  • Euler’s totient function
  • Integers modulo ring
  • Laplacian integral graphs
  • Laplacian matrix

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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