Wiener and Hyper-Wiener Indices of Fusible Graphs of Finite Commutative Rings

Let R be a finite commutative ring with nonzero identity and let GFR denote the fusible graph associated with R. In this paper, we obtain explicit closed-form formulas for the Wiener index and hyper-Wiener index of GFR. Our approach relies on the structure of fusible elements together with the fact that diamGFR≤2, which holds for every finite commutative ring. The obtained formulas are given in terms of the parameters R, FR, and N=x∈R:2x∈FR. For finite Artin rings, these formulas simplify further using the identity FR=R∖NilR. We further show that, for a finite Artin ring R, the complement of GFR is a disjoint union of complete graphs and complete bipartite graphs indexed by the cosets of NilR; as a consequence, GFR is a complete multipartite graph with explicitly known spectrum whenever 2∈NilR, and its full spectrum is also determined when R is odd. We also investigate structural and spectral properties of fusible graphs, including regularity and spectral radius bounds. In particular, we characterize the equality case for the spectral radius in terms of regularity of GFR. Finally, a comparison with the corresponding indices of unit graphs GUR shows that WGFR≤WGUR and HWGFR≤HWGUR whenever both quantities are defined, reflecting the stronger connectivity properties of fusible graphs.