TY - JOUR

T1 - On Laplacian integrability of comaximal graphs of commutative rings

AU - Rather, Bilal Ahmad

AU - Aouchiche, Mustapha

AU - Imran, Muhammed

N1 - Publisher Copyright:
© 2023, The Indian National Science Academy.

PY - 2023

Y1 - 2023

N2 - 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)].

AB - 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)].

KW - Algebraic connectivity

KW - Comaximal graphs

KW - Euler’s totient function

KW - Integers modulo ring

KW - Laplacian integral graphs

KW - Laplacian matrix

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U2 - 10.1007/s13226-023-00364-8

DO - 10.1007/s13226-023-00364-8

M3 - Article

AN - SCOPUS:85145952076

SN - 0019-5588

JO - Indian Journal of Pure and Applied Mathematics

JF - Indian Journal of Pure and Applied Mathematics

ER -