## Abstract

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(Z_{n}) and show that the algebraic connectivity of Γ(Z_{n}) 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(Z_{n}) is an integer if and only if n=p^{α}q^{β}, and hence Γ(Z_{n}) is Laplacian integral if and only if n=p^{α}q^{β}, 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 Z_{n}, Special Matrices, (2022)].

Original language | English |
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Pages (from-to) | 310-324 |

Number of pages | 15 |

Journal | Indian Journal of Pure and Applied Mathematics |

Volume | 55 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 2024 |

## Keywords

- 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