Abstract
For a commutative ring R, the comaximal graph \( \Gamma (R) \) of R is a simple graph with vertex set R and two distinct vertices u and v of \( \Gamma (R) \) are adjacent if and only if \( aR+bR=R \). In this article, we find the Laplacian eigenvalues of \( \Gamma (\mathbb {Z}_{n}) \) and show that the algebraic connectivity of \( \Gamma (\mathbb {Z}_{n}) \) is always an even integer and equals \( \phi (n) \), thereby giving a large family of graphs with integral algebraic connectivity. Further, we prove that the second largest Laplacian eigenvalue of \( \Gamma (\mathbb {Z}_{n}) \) is an integer if and only if \( n=p^{\alpha }q^{\beta },\) and hence \( \Gamma (\mathbb {Z}_{n}) \) is Laplacian integral if and only if \( n=p^{\alpha }q^{\beta },\) where p, q are primes and \( \alpha , \beta \) are non-negative integers. This answers a problem posed by [Banerjee, Laplacian spectra of comaximal graphs of the ring \( \mathbb {Z}_{n} \), Special Matrices, (2022)].
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Rather, B.A., Aouchiche, M. & Imran, M. On Laplacian integrability of comaximal graphs of commutative rings. Indian J Pure Appl Math 55, 310–324 (2024). https://doi.org/10.1007/s13226-023-00364-8
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DOI: https://doi.org/10.1007/s13226-023-00364-8
Keywords
- Laplacian matrix
- Algebraic connectivity
- Comaximal graphs
- Integers modulo ring
- Laplacian integral graphs
- Euler’s totient function