Properties of Idealization Graph Over a Ring

We define the new definition is called the idealization graph. Also, we study some properties of the idealization graph such that the girth is 3 and diameter is equal 2 and study the idealization graph is not Eulerian graph and is a divisor graph. And we show that the idealization graph is a Hamiltonian graph if \(\left| {ann\left( S \right)} \right| = \left| {F\backslash ann\left( S \right)} \right|\) . Further, we determine the Wiener index of the idealization graph, denoted as W (L (+) S)), which quantifies the sum of distances between all pairs of vertices. The Wiener index is calculated as (|ann(S)||S|+2|F\ann(S)||S|−2)|F\ann(S)||S|+(|F||S|−1)|ann(S)||S|. Lastly, we investigate the planarity of the idealization graph and identify conditions for it to be a Planar or One-Planar graph. Specifically, we find that when the cardinality of S is less than or equal to 3 and the annihilator ann(S) is equal to 0.