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 -