On the Spectra and Wiener Index of the Comaximal Graph of a Commutative Ring

Let R be a commutative ring. The comaximal graph Γ(R) is defined as an undirected simple graph whose vertices correspond to the elements of R, where two distinct vertices x and y are adjacent if and only if Rx+Ry=R, with Rx denoting the ideal generated by x. For the case R=Zn, the ring of integers modulo n, the comaximal graph is represented as a generalized composition of appropriately chosen graphs. Building on this concept, we demonstrate that the comaximal graph Γ(R) of any commutative Artinian ring with unity can be expressed as an H-join of the complete and null graphs. Utilizing this structural representation, we derive an explicit formula for the Wiener index of the comaximal graph associated with a commutative Artinian ring. Finally, we determine the spectrum of the comaximal graph for such rings.