Sombor index, polynomial, and energy of $ m $-graphs of finite abelian groups

Let $ H $ be a finite abelian group of order $ n\ge 2 $ and let $ m > 1 $ be an integer. The $ m $-graph $ m $-$ G(H) $ is the simple undirected graph on vertex set $ H $ in which two distinct elements $ a $ and $ b $ are adjacent if and only if $ a^{m} = b $ or $ b^{m} = a $. Building on the structural characterization of connected $ m $-graphs established by Badawi, we investigated degree-based topological invariants of connected $ m $-$ G(H) $ in both the cyclic case $ H\cong\mathbb Z_n $ and the non-cyclic case $ H\cong \mathbb Z_{n_1}\oplus\cdots\oplus\mathbb Z_{n_r}, r\ge 2. $ Using an edge-type decomposition arising from the degree structure of connected $ m $-graphs, we derived explicit formulas for the Sombor index, Sombor polynomial, reduced and modified Sombor indices, Euler Sombor index, average Sombor index, and forgotten topological index. In the non-cyclic setting, the parameter $ \mathcal{G} = \prod_{j = 1}^{r}\gcd(d_{j}, q_{j}) $ governs the edge-type counts $ N_{BS} = D - \mathcal{G} $ and $ N_{AS} = \mathcal{G} - 1 $, which cannot be expressed purely in terms of $ D $ and $ Q $. We also determined closed-form expressions for the Sombor energy in the cases $ Q = 1 $ (star graphs) and $ Q = 2 $ (two-hub trees). All theoretical results were independently verified by exact symbolic computation using SageMath 10.3 and the SymPy library.