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(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 language | English |
---|---|
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