Laplacian spectrum of comaximal graph of the ring ℤn

Abstract

In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) of the ring Zn{{\mathbb{Z}}}_{n} for n>2n\gt 2. We first determine the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for various nn. We show that Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) is Laplacian integral for n=pαqβn={p}^{\alpha }{q}^{\beta }, where p,qp,q are primes and α,β\alpha ,\beta are non-negative integers and hence calculate the number of spanning trees of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for n=pαqβn={p}^{\alpha }{q}^{\beta }. The algebraic and vertex connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) have been shown to be equal for all nn. An upper bound on the second largest Laplacian eigenvalue of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). We then investigate some properties and vertex connectivity of an induced subgraph of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). Some problems have been discussed at the end of this paper for further research.

Keywords

• comaximal graph
• laplacian eigenvalues
• vertex connectivity
• algebraic connectivity
• laplacian spectral radius
• finite ring
• 05c25
• 05c50