Laplacian Spectrum and Vertex Connectivity of the Unit Graph of the Ring Zprqs

In this paper, we examine the interplay between the structural and spectral properties of the unit graph G(Zn) for n=p1r1p2r2...pkrk, where p1,p2,...,pk are distinct primes and k,r1,r2,...,rk are positive integers such that at least one of the ri must be greater than 1. We first analyze the structure of the unit graph of Zn as a generalized join graph under these conditions. We then determine the Laplacian spectrum of G(Zn) and deduce that it is integral for all n. Consequently, we obtain Laplacian spectral radius and algebraic connectivity of G(Zn). We also prove that the vertex connectivity of G(Zpq) is (p−2)q, where 2≠p<q. We deduce the vertex connectivity of G(Zn) when n=prqs, where 2≠p<q are primes and r,s are positive integers. Finally, we present conjectures about the vertex connectivity of G(Zn) when n=p1p2...pk and n=p1r1p2r2...pkrk, where pi are distinct primes, ri are positive integers, and 1≤i≤k.