Laplacian spectrum of comaximal graph of the ring ℤ n

Abstract

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