Analysis of the Wiener Index of Rough Annihilator Graph Over Rough Semirings

An effective analytical and visual tool for comprehending the annihilator relationships inside a rough semiring is its annihilator graph. This paper introduces and investigates rough annihilator graph, denoted R A G ( T ), of the commutative rough semiring T . The vertices of R A G ( T ) are the nonzero rough zero‐divisors and two distinct vertices, R S ( X ) and R S ( Y ), are adjacent iff A n n T ( R S ( X )∇ R S ( Y )) ≠ A n n T ( R S ( X )) ∪ A n n T ( R S ( Y )). The study demonstrates that R A G ( T ) is connected with diameter at most 2 and shows that the rough zero‐divisor graph is a subgraph of rough annihilator graph of the rough semiring. The paper further investigates a new graph called clustered rough annihilator graph, built upon adjacency properties and sharing the same annihilators, where adjacency is defined based on annihilators. The structure and features of the rough annihilator graph are examined in the framework of commutative rough semiring theory. The primary focus of this work is the Wiener index, a distance‐based topological index that measures the structural characteristics of graphs. We pay special attention to the computation for the related algebraic graph constructs. In this paper, we investigate the Wiener index of R A G ( T ), defined as the sum of distances between all pairs of vertices in the graph. We derive explicit expressions for the Wiener index of rough annihilator graph R A G ( T ) and clustered rough annihilator graph C R A G ( T ).